Do the Catholic Church ex cathedra pronouncements about necessity of Catholicism to be saved still apply?

Simply to add to the answer given by Geremia...

It is quite possible to consistently believe that "outside of The [Catholic] Church there is no salvation", and square it away with some of the more inclusive statements that Popes have spoken recently. For consistency's sake one must believe this statement in the same way that Catholics believe the following parallel statement of Jesus Christ: "I am the way, the truth, and the life. No one comes through The Father except through me' (John 14:6).

In fact, given the close connection that Catholic theology makes between Jesus Christ and The Catholic Church, those two statements go hand in hand: the statement about The Church follows because of the statement about Jesus Christ.

Often, the following doubt is raised to Christians about the statement about Jesus: `and what about the aborigine living in the middle of the deepest jungle who has never heard of Christ, but who is a good man? Or what about the Buddhist monk who doesn't believe in Christ, but who is gentle, compassionate, truthful, just, etc.?'

A rough Catholic (Christian?) answer to this doubt is to re-ascertain that the only way to The Father is through Jesus Christ, and that if it should happen that the aborigine or the Buddhist make it to heaven it will be entirely and solely through Jesus Christ. And the only logical way this can be is that Jesus Christ is acting in the lives of the hypothetical aborigine and hypothetical Buddhist who make it to Heaven, even if they themselves are not aware of it.

Indeed, as Catholic theology teaches, God --- always through Jesus --- is the source of goodness, the source of justice, the source of love, the source of truth, etc. More than that, Catholic theology claims that God is Love, God is Justice, God is Goodness, God is Truth, etc. Thus, if people anywhere are binding themselves to truth, justice, goodness, love, etc., they are binding themselves to Jesus. Certainly, if they do not know the fullness of truth, their binding will be imperfect; but nonetheless, God can work through that imperfect obedience to achieve their salvation.

Similarly, because the essence of The Church is Jesus Christ --- The Church is the mystical body of Christ --- to be bound to Jesus is to be bound to The Church. Thus, it is true that "outside the Church there is no salvation" because outside Jesus Christ there is no salvation. However, just as one can be deeply in love with Jesus Christ without fully realizing it, one can be a member of The Church without fully seeming to be.

In fact, as the teachings of the Council of Vatican II reiterated, the way to look at membership in The Church is as overlapping circles... While to be a visible member of The Catholic Church means one can be in full communion with The Church, not being a visible member of The Church means that one is in imperfect communion with The Church, sharing some of Her (and His) teachings, while lacking others, and holding some errors as well. The closer that the circles coincide, the closer to full communion one is. If there be any overlap at all, then, to that measure, one is participating in the life of The Church, even if not visibly.

If I understand your question as Did the Second Vatican Council change the Catholic doctrine on the Church?

William Cardinal Levada, Prefect of the Congregation for the Doctrine of the Faith (CDF), published the following response (sans footnotes) on June 29, 2007:

The Second Vatican Council neither changed nor intended to change this doctrine, rather it developed, deepened and more fully explained it.

This was exactly what John XXIII said at the beginning of the Council. Paul VI affirmed it and commented in the act of promulgating the Constitution Lumen gentium: “There is no better comment to make than to say that this promulgation really changes nothing of the traditional doctrine. What Christ willed, we also will. What was, still is. What the Church has taught down through the centuries, we also teach. In simple terms that which was assumed, is now explicit; that which was uncertain, is now clarified; that which was meditated upon, discussed and sometimes argued over, is now put together in one clear formulation”. The Bishops repeatedly expressed and fulfilled this intention. - Source: Responses to Some Questions Regarding Certain Aspects of the Doctrine on the Church | CDF.

Therefore from the Catholic Church's perspective, what she has always taught about herself and the necessity to belong to her for salvation has never changed and is till valid today as when the Church first taught it [which is from the very beginning].

When a Pope is Infallible

The First Vatican Council's Pastor Æternus said, under Pope Pius IX's authority, regarding papal infallibility:

…we teach and define that it is a dogma divinely revealed: that the Roman Pontiff, when he speaks ex cathedra, that is, when in discharge of the office of pastor and doctor of all Christians, by virtue of his supreme Apostolic authority, he defines a doctrine regarding faith or morals to be held by the universal Church, by the divine assistance promised to him in blessed Peter, is possessed of that infallibility with which the divine Redeemer willed that his Church should be endowed for defining doctrine regarding faith or morals; and that therefore such definitions of the Roman Pontiff are irreformable of themselves, and not from the consent of the Church.

Thus, the following critera must hold for a pope's statement to be ex cathedra:

He must:

1. speak for the whole Church,
2. invoke all his authority,
3. intend to definitively define dogma, and
4. speak regarding the faith and/or morals.

Particular Examples

For example, Pope Eugene IV did meet all these criteria in Cantate Domino (1442):

1. speaks for the whole Church: "The most Holy Roman Church firmly believes, professes and preaches…"
2. invokes all his authority: The document is an authoritative papal bull.
3. intend to definitively define dogma: In this case, he is not defining anything new; he's only relaying what has been said before him (like in Unam Sanctam of 1302) regarding the necessity of the Church for salvation.
4. speak regarding the faith and/or morals: Yes. In this case, he speaks of both.

A similar analysis can be made for Pope Boniface VIII's Unam Sanctam (1302):

1. speaks for the whole Church: Bulls are addressed to the whole Church.
2. invokes all his authority: The document is an authoritative papal bull.
3. intend to definitively define dogma: "…we declare, we proclaim, we define…"
4. speak regarding the faith and/or morals: Yes. He speaks about what is necessary for salvation, which pertains to the faith.

Francis's statements do not meet these criteria.

1. speaks for the whole Church: This is unclear. Was he giving the homily only for those present in the Wednesday audience, or the whole Church? Certainly, modern media is able to make everything coming out of a pope's mouth sound like he's speaking for the whole Church.
2. invokes all his authority: Francis nowhere uses solemn language like, for example, Pope Pius XII used to declare the dogma of the Assumption of Mary into heaven in the Apostolic Constitution Munificentissimus Deus (1950):

For which reason, after we have poured forth prayers of supplication again and again to God, and have invoked the light of the Spirit of Truth, for the glory of Almighty God who has lavished his special affection upon the Virgin Mary, for the honor of her Son, the immortal King of the Ages and the Victor over sin and death, for the increase of the glory of that same august Mother, and for the joy and exultation of the entire Church; by the authority of our Lord Jesus Christ, of the Blessed Apostles Peter and Paul, and by our own authority, we pronounce, declare, and define it to be a divinely revealed dogma:

that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul into heavenly glory. 3. intend to definitively define dogma: Homilies have never been used to define dogma, nor did he express his intent to define dogma. 4. speak regarding the faith and/or morals: Yes, he was speaking about redemption.

Also, John Paul II's Catechism of the Catholic Church is fallible because it contradicts previous Church teaching on some points. See this for an in-depth analysis.

Whether a Pope Making Erroneous or Heretical Statements is Compatible with the Church's Sanctity

The 4 marks of the Church are:

1. unity
2. sanctity
3. Catholic (universality)
4. apostolic

The first mark, unity, does not just mean a unity of the members of the Church today. It means there must be a continuity in the Church's teachings and dogma from the time Christ founded the Church until today.

The second mark of the Church, its sanctity, means the Church cannot teach error or lead one into error.

What about when a bishop does teach error or heresy? For example, Pope Honorius I is thought to have taught the monothelite heresy (that Christ only has one will) in a private letter (see this for the controversy), and St. Frances de Sales, in his The Catholic Controversy, considered the possibility that he was a formal heretic and, since heretics are outside the Church, thus also an anti-pope. It's worth quoting the beginning of The Catholic Controversy (pp. 305 f.), the chapter on "how ministers have violated their authority", which teaching influenced the First Vatican Council's definition on papal authority quoted above:

Under the ancient law the High Priest did not wear the Rational except when he was vested in the pontifical robes and was entering before the Lord. Thus we do not say that the Pope cannot err in his private opinions, as did John XXII.; or be altogether a heretic as perhaps Honorius was. Now when he is explicitly a heretic, he falls ipso facto from his dignity and out of the Church, and the Church must either deprive him, or, as some say, declare him deprived, of his Apostolic See, and must say as S. Peter did : Let another take his bishopric. [Acts 1] When he errs in his private opinion he must be instructed, advised, convinced; as happened with John XXII, who was so far from dying obstinate or from determining anything during his life concerning his opinion, that he died whilst he was making the examination which is necessary for determining in a matter of faith, as his successor [Pope Benedict XII] declared in the Extravagantes which begins Benedictus Deus. But when he is clothed with the pontifical garments, I mean when he teaches the whole Church as shepherd, in general matters of faith and morals, then there is nothing but doctrine and truth. And in fact everything a king says is not a law or an edict, but that only which a king says as king and as a legislator. So everything the Pope says is not canon law or of legal obligation; he must mean to define and to lay down the law for the sheep, and he must keep the due order and form.

Also, the following passage from St. Vincent Lerins's The Commonitory is worth quoting in full, too; it gives the rules for how to deal with a bishop or priest teaching error or heresy: stick to tradition.

Also in the Catholic Church itself we take great care that we hold that which has been believed everywhere, always, by all. For that is truly and properly Catholic, as the very force and meaning of the word shows, which comprehends everything almost universally. And we shall observe this rule if we follow universality, antiquity, consent. We shall follow universality if we confess that one Faith to be true which the whole Church throughout the world confesses; antiquity if we in no wise depart from those interpretations which it is plain that our ancestors and fathers proclaimed; consent if in antiquity itself we eagerly follow the definitions and beliefs of all, or certainly nearly all, priests and doctors alike.

What, then, will the Catholic Christian do if any part of the Church has cut itself off from the communion of the Universal Faith? What surely but prefer the soundness of the whole body to a pestilent and corrupt member?

What if some novel contagion seek to infect the whole Church, and not merely a small portion of it? Then he will take care to cling to antiquity, which cannot now be led astray by any novel deceit.

What if in antiquity itself error be detected on the part of two or three men, or perhaps of a city, or even of a province? Then he will look to it that he prefer the decrees of an ancient General Council, if such there be, to the rashness and ignorance of a few.

But what if some error spring up concerning which nothing of this kind is to be found? Then he must take pains to find out and compare the opinions of the ancients, provided, of course, that such remained in the communion and faith of the One Catholic Church, although they lived in different times and places, conspicuous and approved teachers; and whatever he shall find to have been held, written and taught, not by one or two only, but by all equally and with one consent, openly, frequently and persistently, that he must understand is to be believed by himself also without the slightest hesitation.

The Sedevacantist vs. Sedeplenist Debate in the Church Today

Sedevacantists (from the Latin sede vacante = "vacant chair") are Catholics who believe currently there is no pope. They generally consider John XXIII, Paul VI, John Paul I, John Paul II, Benedict XVI, and Francis all anti-popes. They argue that since these men have proclaimed error and heresy in an official manner, they, as St. Francis de Sales said, ipso facto fall out of the Church. (St. Robert Bellarmine, doctor of the Church and "Master of Controversies," says the same.) What is no longer part of the body of the Church cannot be its head; thus, they are not real popes.

Sedeplenists argue that they are true popes. Although they sometimes agree these pope claimants teach error and heresy, sedeplenists do not consider them formal heretics because they judge them as lacking pertinacity in proclaiming error and heresy or as being invincibly ignorant. They say only a future pope or council can depose someone who could be an anti-pope.

There is also the material pope thesis, which essentially says these pope claimants are more like kings than popes.

The Church's Theological Notes or Qualifications

Listed below are the so-called theological notes and their associated censures from the table in Sixtus Cartechini, S.J.'s 1951 work De Valore Notarum Theologicarum (On the Value of the Theological Notes), which confessors have used when dealing with erudite penitents. (It's also available in Italian translation.) The theological notes are a way of classifying the proximity of a theological proposition to revelation. (For a good history of the development of these notes, see The development of the theological censures after the Council of Trent: (1563-1709) by John Cahill, O.P.)

There is only room for "reasonable interpretative variation" in the lowest of the notes (#9 and #10). As Pope Pius XII wrote in Humani Generis:

Popes generally leave theologians free in those matters which are disputed in various ways by men of very high authority in this field; but history teaches that many matters that formerly were open to discussion [e.g., the Immaculate Conception], no longer now admit of discussion.

For the Immaculate Conception, the discussion ceased when Bl. Pope Pius IX defined it dogma in Ineffabilis Deus (1852).

1. Theological note: Dogma.
   Equivalent terms: Dogma of faith; de fide, de fide Catholica; de fide divina et Catholica.
   Explanation: A truth proposed by the Church as revealed by God.
   Examples: The Immaculate Conception; all the contents of the Athanasian Creed.
   Censure attached to contradictory proposition: Heresy
   Effects of denial: Mortal sin committed directly against the virtue of faith, and, if the heresy is outwardly professed, excommunication is automatically incurred and membership of the Church forfeited.
   Remarks: A dogma can be proposed either by a solemn definition of pope or council, or by the Ordinary Magisterium, as in the case of the Athanasian Creed, to which the church has manifested her solemn commitment by its long-standing liturgical and practical use and commendation.
2. Theological Note: Doctrine of ecclesiastical faith
   Equivalent term: De fide ecclesiastica definita
   Explanation: A truth not directly revealed by God but closely connected with Divine revelation and infallibly proposed by the Magisterium.
   Example: The lawfulness of communion under one kind.
   Censure attached to contradictory proposition: Heresy against ecclesiastical faith.
   Effects of denial: Mortal sin directly against faith, and, if publicly professed, automatic excommunication and forfeiture of membership of Church.
   Remarks: It is a dogma that the Church's infallibility extends to truths in this sphere, so one who denies them denies implicitly a dogma or Divine faith.
3. Theological Note: Truth of Divine faith.
   Equivalent term: De fide divina.
   Explanation: A truth revealed by God but not certainly proposed as such by the Church.
   Example: Christ claimed from the beginning of His public life to be the Messias.
   Censure attached to contradictory proposition: Error (in faith).
   Effects of denial: Mortal sin directly against faith, but no loss of Church membership. May incur a canonical penalty.
4. Theological Note: Proximate to faith.
   Explanation: A doctrine all but unanimously held as revealed by God.
   Example: Christ possessed the Beatific Vision throughout his life on earth.
   Censure attached to contradictory proposition: Proximate to error.
   Effects of denial: Mortal sin indirectly against faith.
5. Theological Note: Theologically certain.
   Equivalent term: Dogmatic fact; theological conclusion.
   Explanation: A truth logically following from one proposition which is Divinely revealed and another which is historically certain.
   Example: Legitimacy of Pope Pius XI.
   Censure attached to contradictory proposition: Error (in theology).
   Effects of denial: Mortal sin against faith.
6. Theological Note: Catholic doctrine.
   Equivalent term: Catholic teaching.
   Explanation: A truth authentically taught by the Ordinary Magisterium but not as revealed or intimately connected with revelation.
   Example: Invalidity of Anglican Orders; validity of Baptism conferred by heretic or Jews.
   Censure attached to contradictory proposition: Temerarious.
   Effects of denial: Mortal sin indirectly against faith.
   Remarks: The expression Catholic doctrine is sometimes applied to truths of a higher order also, but never of a lower one. In some cases the appropriate censure may be graver than "temerarious".
7. Theological Note: Certain.
   Equivalent term: Common; theologically certain.
   Explanation: A truth unanimously held by all schools of theologians which is derived from revealed truth, but by more than one step of reasoning.
   Example: The true and strict causality of the sacraments.
   Censure attached to contradictory proposition: Temerarious.
   Effects of denial: Usually, mortal sin of temerity.
   Remarks: Proportionately grave reason can sometimes justify an individual who has carefully studied the evidence in dissenting from such a proposition; since it is not completely impossible for all the theological schools to err on such a matter, although it would be highly unusual and contrary to an extremely weighty presumption.
8. Theological Note: Safe.
   Explanation: Affirmed in doctrinal decrees of Roman Congregations.
   Example: That Christ will not reign visibly on earth for a thousand years after Antichrist.
   Censure attached to contradictory proposition: Unsafe/temerarious.
   Effects of denial: Mortal sin of disobedience and perhaps imprudence.
   Remarks: Exterior assent is absolutely required and interior assent is normally required, since, though not infallible, the Congregations possess true doctrinal authority and the protective guidance of the Holy Ghost.
9. Theological Note: Very common/commoner.
   Explanation: The most solidly founded or best attested theological opinion on a disputed subject.
   Example: Antichrist will be of the tribe of Dan.
   Censure attached to contradictory proposition: None.
   Effects of denial: None.
   Remarks: Very common or commoner opinions can be mistaken and there is no obligation to follow them though prudence inclines us to favour them as a general policy. It should be noted that an opinion which is "very common" is less well established than one which is "common" which implies moral unanimity of theological schools.
10. Theological Note: Probable.
    Explanation: A theological opinion which is well founded either on the grounds of its intrinsic coherence or the extrinsic weight of authority favouring it.
    Example: Judas received Holy Communion at the Last Supper. Judas did not receive Holy Communion at the Last Supper.
    Censure attached to contradictory proposition: None.
    Effects of denial: None.
    Remarks: The better founded of two conflicting opinions is referred to as more probable; but Catholics are free to prefer some other opinion for any good reason.