What major problems does Luther have with the Roman Catholic Church and how does he make his points?

Martin Luther never wanted to be a monk in the first place. He didn't want to live a life of celibacy, either. He only became a monk because he got caught in a really bad storm and bargained with St. Anne that he'd be a monk if he survived the storm. That tells you something about the man right there. He wasn't called and didn't want to do it.

Once a monk, he Is said to have driven the other priests crazy because he went to confession so much. He had a guilt complex from his childhood. One account said that to give the priests in the monastery a break, he was sent away from the monastery to teach the Bible to members of a Parish. It was there that he misinterpreted the passage in which Paul was encouraging the Jewish Christians to be kind and non-judgmental to the gentile converts who did not follow the laws of Moses known as works.
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Ironically, this priest who didn't want to be a priest and wasn't called went from going to confession so often that the other priests sent him away to deciding that it wasn't necessary to confess at all. Luther's problems with the church were mostly centered around confession and indulgences.

As for indulgences, the church never sold them. Father Mike explains.

https://www.youtube.com/watch?v=HpdUZ3kWjM8



https://www.youtube.com/watch?v=1o8oIELbNxE

https://www.youtube.com/watch?v=Z92-glVHyCw

Main points from Luther’s 95 Theses:
6. The pope himself cannot remit guilt, but only declare and confirm that it has been remitted by God; or, at most, he can remit it in cases reserved to his discretion. Except for these cases, the guilt remains untouched.
13. Death puts an end to all the claims of the Church; even the dying are already dead to the canon laws, and are no longer bound by them.
21. Hence those who preach indulgences are in error when they say that a man is absolved and saved from every penalty by the pope's indulgences.
36. Any Christian whatsoever, who is truly repentant, enjoys plenary remission from penalty and guilt, and this is given him without letters of indulgence.
37. Any true Christian whatsoever, living or dead, participates in all the benefits of Christ and the Church; and this participation is granted to him by God without letters of indulgence.
42. Christians should be taught that the pope does not at all intend that the purchase of indulgences should be understood as at all comparable with the works of mercy.
43. Christians should be taught that one who gives to the poor, or lends to the needy, does a better action than if he purchases indulgences.
44. Because, by works of love, love grows and a man becomes a better man; whereas, by indulgences, he does not become a better man, but only escapes certain penalties.
66. The treasures of the indulgences are the nets which to-day they use to fish for the wealth of men.
67. The indulgences, which the merchants extol as the greatest of favours, are seen to be, in fact, a favourite means for money-getting.
68. Nevertheless, they are not to be compared with the grace of God and the compassion shown in the Cross.
79. It is blasphemy to say that the insignia of the cross with the papal arms are of equal value to the cross on which Christ died.
86. Again: since the pope's income to-day is larger than that of the wealthiest of wealthy men, why does he not build this one church of St. Peter with his own money, rather than with the money of indigent believers?

Do your own homework.

Look up Luther's 95 Theses which he nailed to a church door in Germany. It's available online, and self-explanatory.

So

Sleeps with infants