What, exactly, happened to the good thief on the cross, and why? (Catholic perspective)

Much of the Question has already been answered, so I will address the concern of the OP.

“It would seem that he did no or minimal good works, and it's most likely that he wasn't baptised.”

As to his Baptism:

The CCC defines Martyr as fallows

MARTYR: A witness to the truth of the faith, in which the martyr endures even death to be faithful to Christ. Those who die for the faith before having received Baptism are said to have received a “baptism of blood,” by which their sins are forgiven and they share in the death and Resurrection of Christ

The fallowing passages detail more the benefits to the Martyr as it pertains to Baptism.

CCC 2473 Martyrdom is the supreme witness given to the truth of the faith: it means bearing witness even unto death. The martyr bears witness to Christ who died and rose, to whom he is united by charity. He bears witness to the truth of the faith and of Christian doctrine. He endures death through an act of fortitude. “Let me become the food of the beasts, through whom it will be given me to reach God.”

CCC 1258 The Church has always held the firm conviction that those who suffer death for the sake of the faith without having received Baptism are baptized by their death for and with Christ. This Baptism of blood, like the desire for Baptism, brings about the fruits of Baptism without being a sacrament. (2473)

CCC 852 Missionary paths. The Holy Spirit is the protagonist, “the principal agent of the whole of the Church’s mission.” It is he who leads the Church on her missionary paths. “This mission continues and, in the course of history, unfolds the mission of Christ, who was sent to evangelize the poor; so the Church, urged on by the Spirit of Christ, must walk the road Christ himself walked, a way of poverty and obedience, of service and self-sacrifice even to death, a death from which he emerged victorious by his resurrection.” So it is that “the blood of martyrs is the seed of Christians.”

As to His Works:

Now, as every Good Bible Believing Protestant will tell you, the only thing the Good Thief on the did was have faith and he was saved. Well, if that was true I might be a Protestant.

Luke 23=39-43

One of the criminals who were hanged there kept deriding him and saying, “Are you not the Messiah? Save yourself and us!” But the other rebuked him, saying, “Do you not fear God, since you are under the same sentence of condemnation? And we indeed have been condemned justly, for we are getting what we deserve for our deeds, but this man has done nothing wrong.” Then he said, “Jesus, remember me when you come into your kingdom.” He replied, “Truly I tell you, today you will be with me in Paradise.”

The passage above is the interaction between the Good thief and our Lord, but starts with the ridicule of the bad thief.

"Are you not the Messiah? Save Yourself and Us!"

This behavior was prophesied in the Old Testament, already experienced. See Matthew 27:40, Hebrews 2:8 as it refers to the Temptation by Satan in the desert.

“Do you not fear God, since you are under the same sentence of condemnation"?

The words of the God thief, chastising his fellow robber on the Cross, defending the Dying Christ as he suffers with Christ on the Cross. The effort it must have taken to say these words under such conditions. While dying, he uses his last moments in life to stand up for a suffering man, a man suffering just like himself.

"And we indeed have been condemned justly, for we are getting what we deserve for our deeds"

He confesses his sins and accepts his punishment as just for his Crimes: Reconciliation and Repentance said out loud and publicly.

"but this man has done nothing wrong.”

The Thief now recognizes the sinless bridegroom, Proclaims his innocence not only to the Bad thief but to all those who are at Calvary: a public recognition of him being the spotless lamb.

“Jesus, remember me when you come into your kingdom.”

The thief's final statement, the recognition that Christ is King, but not of this world, but the world to come, and that Christ has the power to save him. Can you imagine, the Holy Spirit Giving Christ such a gift even as he is suffering and dying, recognition of his Kingdom.

Faith Alone? I think not. He suffered with the Christ, in the same manner that Christ himself suffered; he didn't do it silently or without works, he did it in such a way that his very actions saved him. His works were many, and his baptism was the same as Christ's Baptism unto death.