According to Catholicism, in what cases is Purgatory necessary?

There are at least two very poignant biblical principles concerning purgatory

Judas rallied his army and went to the city of Adullam. As the seventh day was approaching, they purified themselves according to custom and kept the sabbath there. On the following day, since the task had now become urgent, Judas and his companions went to gather up the bodies of the fallen and bury them with their kindred in their ancestral tombs. But under the tunic of each of the dead they found amulets sacred to the idols of Jamnia, which the law forbids the Jews to wear. So it was clear to all that this was why these men had fallen. They all therefore praised the ways of the Lord, the just judge who brings to light the things that are hidden.

Turning to supplication, they prayed that the sinful deed might be fully blotted out. The noble Judas exhorted the people to keep themselves free from sin, for they had seen with their own eyes what had happened because of the sin of those who had fallen. He then took up a collection among all his soldiers, amounting to two thousand silver drachmas, which he sent to Jerusalem to provide for an expiatory sacrifice. In doing this he acted in a very excellent and noble way, inasmuch as he had the resurrection in mind; for if he were not expecting the fallen to rise again, it would have been superfluous and foolish to pray for the dead. But if he did this with a view to the splendid reward that awaits those who had gone to rest in godliness, it was a holy and pious thought. Thus he made atonement for the dead that they might be absolved from their sin.
2 Maccabees 12:38-46 - NABRE

I quote this in full because the explanation is at least as important as the fact. The fact is, whoever wrote 2 Maccabees was thinking the same thing that lots of Protestants think, It is superflous and foolish to pray for the dead. The only way that these fallen away Jews would "rest in godliness" is to be atoned for their sins.

Matthew 5:26 and Luke 12:59 shed further light on how deep the debt goes that those who die in their sins need to repay. And the guy in Matthew 18:34 who weaseled his way out of his own debt only to not forgive another man his debt got what the commentary in the NABRE says is "an endless debt". That's not so good.

My diocese's bishop likes to joke about feeling fortunate that St. Peter will ask him to turn the lights off when they kick everyone out of purgatory on the last day, but that's still infinitely better than Hell. Purgatory doesn't even make the list on the Catechesis of "Last Things" (Death, Judgment, Heaven and Hell) and St. John Paul the Great more or less relegated Purgatory to a state. Not a plane of existence, but a state of the soul. Purgatory can't have any meaning after the last judgment, so what my bishop was saying was, he's not banking on making it into heaven until everyone else is already there eating breakfast.

I only bring all this up to clarify that Heaven is different than purgatory and the post-resurrection heaven must be a place because our bodies will be there reunited with our souls. We used to refer to Purgatory as the "Church Suffering" but that made a lot of people frown when they thought of their not-too-saintly aunts and uncles. So now we call them the "Holy souls in purgatory" but the point is, we don't call 'em the holy souls and bodies in purgatory. And that's the real point, just like Moses and the Israelites who were pretty convinced that you'd die if you saw God face to face. The souls in purgatory are there because they literally can't do what the souls in heaven (the Church Triumphant) are capable of doing; seeing God's glory, the beatific vision.

A person's action for good or for evil change a person so that a person is good or evil [as evidenced in life] and thereby acceptable or not acceptable to God.

This is the biblical language of St. John the Baptist referencing good wheat vs. chaff. The former is collected in God's barn [heaven] and the latter burnt in unquenchable fire [hell]. cf. Mt 3:12

There is an in between stage and the better analogy is impure gold. For that a purifying fire is required so that only pure gold enters into God's presence. If that purification is not completed in this world, it is accomplished after departure from it.

This is supported by the following Biblical passages:

If any man’s work is burned up, he will suffer loss, though he himself will be saved, but only as through fire.[1 Cor 3:15]

and the judge to the guard, and you be put in prison; truly, I say to you, you will never get out till you have paid the last penny.[Mt 5: 25-26]

Hell is for ever. Purgatory is the prison one can get out of after paying the last penny/being purified. In Catholicism, those pennies, the things for which purification is required, are: venial sins [displeasing to God but not grave enough to cause death in the soul as mortal sins do] and the unpaid debt of temporal punishment due to those sins of which the guilt has been forgiven.

cf. Purgatory | PENNY CATECHISM, 106-109

And

CCC 1472 To understand this doctrine and practice of the Church, it is necessary to understand that sin has a double consequence. Grave sin deprives us of communion with God and therefore makes us incapable of eternal life, the privation of which is called the "eternal punishment" of sin. On the other hand every sin, even venial, entails an unhealthy attachment to creatures, which must be purified either here on earth, or after death in the state called Purgatory. This purification frees one from what is called the "temporal punishment" of sin. These two punishments must not be conceived of as a kind of vengeance inflicted by God from without, but as following from the very nature of sin. A conversion which proceeds from a fervent charity can attain the complete purification of the sinner in such a way that no punishment would remain.¹

^{1. cf. Council of Trent (1551): DS 1712-1713; (1563): 1820.}

From the last sentence, we gather how the 'good thief' was able to be in paradise with LORD on that very same day as the LORD promised him.

Very good question that I would like to see if I can answer to your satisfaction or at least give you some pointers to where to obtain a deeper understanding of the mystery of what happens to our soul after death. You can find biblical passages in my references that support this understanding by the Church and has been expanded and deepened over the centuries, but nevertheless, for sure, nobody knows, only God.

Purgatory is not only necessary for the reasons you list but also is necessary, but not limited to, erasing from our soul, not the sin per se, but the tendency for us to sin. Since we would go to purgatory "after" our personal judgement with Christ at which point we would have accepted Him or rejected Him.

This is call "Concupiscence." Here is an extract of the New Advent Catholic Encyclopedia under this subject:

Christ by His death redeemed mankind from sin and its bondage. In baptism the guilt of original sin is wiped out and the soul is cleansed and justified again by the infusion of sanctifying grace. But freedom from concupiscence is not restored to man, any more than immortality; abundant grace, however, is given him, by which he may obtain the victory over rebellious sense and deserve life everlasting.

For an easy way to envision this, take this example: how come, even though Christ has forgiven us, in the case of Catholics via the Sacrament of Reconciliation or Confession or in protestant circles, by Accepting Christ as my Savior and be saved, we still sin and do bad things? It is very, very hard to work on our bad behaviors and tendencies and we see we fall again and again. I am talking here about the true at heart and honest to themselves believers, not the people that say one thing and do another or never had the intention to amend their ways.

So, purgatory is where we work away, with the Grace of God, that last remaining tendency to not do the will of God freely and instantly, with no second thoughts of what we want or what makes us feel good. It is the point where there is no distinction between our will and God's Will.

In the case of the Good Thief on the Cross, we do not know what that infusion of Grace my have been to participate and witness the agony and sacrifice of Jesus on the Cross. He may have been transformed so deeply by the experience that when Christ says to him "Today you will be with me in paradise." (Luke 23:43) his passage through Purgatory may have been just an instant or a few hours. Then again, we don't know what a day is (or the sense of today) in the language of eternity.

So, with God's infinity Mercy and Love, he has given way to ensure we have all that we would need in order stand in front of him in His Kingdom with a soul as pure and white as snow to praise and glorify him through eternity!

God bless!

Reference

Catechism of the Catholic Church 1031