Does the Catholic Church officially recognize Protestants as Christians?

“Do Catholics officially recognize Protestants as Christians?" More generally: Are baptized non-Catholics Christians?

The answer is no.

A Christian is someone who is a member of the Mystical Body of Christ. Baptized non-Catholics are not members of the Church because, as Pope Pius XII wrote in his encyclical Mystici Corporis Christi §22:

Actually only those are to be included as members of the Church who have

1. been baptized

and

2. profess the true faith,*

and

3. who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed.

“For in one spirit” says the Apostle, “were we all baptized into one Body, whether Jews or Gentiles, whether bond or free.” As therefore in the true Christian community there is only one Body, one Spirit, one Lord, and one Baptism, so there can be only one faith. And therefore if a man refuse to hear the Church let him be considered — so the Lord commands — as a heathen and a publican. [cf. Mt. 18:17] It follows that those [who] are divided in faith or government cannot be living in the unity of such a Body, nor can they be living the life of its one Divine Spirit.
_{*[in its entirely; cf. Satis Cognitum]}

Protestants are at least material heretics; they do not hold the same faith as Catholics. They deny the papacy, a truth necessary for being a member of the Church. They do not even recognize the visible Church (its government). And, as Pope Leo XIII said in Satis Cognitum, an encyclical on the unity of the Church consisting in being “one in faith, in government, and in communion:”

The practice of the Church has always been the same, as is shown by the unanimous teaching of the Fathers, who were wont to hold as outside Catholic communion, and alien to the Church, whoever would recede in the least degree from any point of doctrine proposed by her authoritative Magisterium.

Thus, Protestants are not Christian.

Addressing your question of salvation: Is

my salvation as a whole, outside the Catholic church,…recognized by my trust in Christ?

No, the Catholic Church, in the Council of Trent session VI, condemns the Protestant notion of sola fide ("saved by faith alone"):

no one ought to flatter himself up with faith alone (sola fide), fancying that by faith alone (sola fide) he is made an heir [i.e., a member of the Church.]

---

CANON IX.—If any one saith, that by faith alone (sola fide) the impious is justified, in such wise as to mean, that nothing else is required to cooperate in order to the obtaining the grace of Justification, and that it is not in any way necessary, that he be prepared and disposed by the movement of his own will: let him be anathema.

I have always thought that the Pope did not recognize the validity of the faith of those outside of the Catholic church.

Yes, that's true—for example, Pope Eugene IV with the Council of Florence solemnly defined the following infallible (and thus unchangeable) dogma in Cantate Domino:

The sacrosanct Roman Church, founded by the voice of our Lord and Savior…firmly believes, professes, and proclaims that those not living within the Catholic Church, not only pagans, but also Jews and heretics and schismatics cannot become participants in eternal life, but will depart “into everlasting fire which was prepared for the devil and his angels” [Matt. 25:41], unless before the end of life the same have been added to the flock; and that the unity of the ecclesiastical body is so strong that only to those remaining in it are the sacraments of the Church of benefit for salvation, and do fastings, almsgiving, and other functions of piety and exercises of Christian service produce eternal reward, and that no one, whatever almsgiving he has practiced, even if he has shed blood for the name of Christ, can be saved, unless he has remained in the bosom and unity of the Catholic Church.

_{A good book on this topic is Msgr. Joseph Clifford Fenton's The Catholic Church and Salvation: In the Light of Recent Pronouncements of the Holy See.}

According to Pope Benedict XV's encyclical Ad Beatissimi Apostolorum, a Christian is a Catholic who holds the faith in its entirety:

24. … Such is the nature of Catholicism that it does not admit of more or less, but must be held as a whole or as a whole rejected: "This is the Catholic faith, which unless a man believe faithfully and firmly; he cannot be saved" (Athanas. Creed). There is no need of adding any qualifying terms to the profession of Catholicism: it is quite enough for each one to proclaim "Christian is my name and Catholic my surname"

“Do Catholics officially recognize Protestants as Christians?" More generally: Are baptized non-Catholics Christians? No.

A Christian is someone who is a member of the Mystical Body of Christ. Baptized non-Catholics are not members of the Church because, as Pope Pius XII wrote in his encyclical Mystici Corporis Christi §22:

Actually only those are to be included as members of the Church who have

1. been baptized

and

2. profess the true faith,*

and

3. who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed.

“For in one spirit” says the Apostle, “were we all baptized into one Body, whether Jews or Gentiles, whether bond or free.” As therefore in the true Christian community there is only one Body, one Spirit, one Lord, and one Baptism, so there can be only one faith. And therefore if a man refuse to hear the Church let him be considered — so the Lord commands — as a heathen and a publican. [cf. Mt. 18:17] It follows that those [who] are divided in faith or government cannot be living in the unity of such a Body, nor can they be living the life of its one Divine Spirit.
_{*[in its entirely; cf. Satis Cognitum]}

Protestants are at least material heretics; they do not hold the same faith as Catholics. They deny the papacy, a truth necessary for being a member of the Church. They do not even recognize the visible Church (its government). And, as Pope Leo XIII said in Satis Cognitum, an encyclical on the unity of the Church consisting in being “one in faith, in government, and in communion:”

The practice of the Church has always been the same, as is shown by the unanimous teaching of the Fathers, who were wont to hold as outside Catholic communion, and alien to the Church, whoever would recede in the least degree from any point of doctrine proposed by her authoritative Magisterium.

Thus, Protestants are not Christian.

That's funny to see that we all lack of information (see my answer which asked exactly the opposite, that is to say if some protestant church believed that Catholics were pagans).

In any case, reading the already mentioned CCC 838 and also other paragraphs, it's quite clear that Catholic church believes that everybody can find salvation if their heart sincerely searches for God

Those who, through no fault of their own, do not know the Gospel of Christ or his Church, but who nevertheless seek God with a sincere heart, and, moved by grace, try in their actions to do his will as they know it through the dictates of their conscience - those too may achieve eternal salvation. (CCC 847)

Hence if through your faith you sincerely search for God, you are saved.

Be careful to some traps that we might fall in, like the one raised by the sentence

"Outside the Church there is no salvation"

However, CCC 846 reveals the answer:

846 How are we to understand this affirmation, often repeated by the Church Fathers? Re-formulated positively, it means that all salvation comes from Christ the Head through the Church which is his Body:

Basing itself on Scripture and Tradition, the Council teaches that the Church, a pilgrim now on earth, is necessary for salvation: the one Christ is the mediator and the way of salvation; he is present to us in his body which is the Church. He himself explicitly asserted the necessity of faith and Baptism, and thereby affirmed at the same time the necessity of the Church which men enter through Baptism as through a door. Hence they could not be saved who, knowing that the Catholic Church was founded as necessary by God through Christ, would refuse either to enter it or to remain in it.

The last, bolded sentence, includes respect of God for our personality, thoughts, mentality. So if in your conscience you do not believe that the Catholic church is "founded as necessary by God through Christ", then you are still in the position of searching God with a sincere Heart.