How can one atone for a vow to practice masturbation repeatedly for ten years, and is it obligatory to fast for two months or feed sixty poor people, and how are missed prayers from years ago due to sleep or forgetfulness to be made up?

All praise is due to Allah for guiding the questioner to abandon the bad habit. As for the vow mentioned, it is a vow of contention (Nathr al-Lajaj), where the one who made it has the choice between fulfilling the vow or offering an expiation for an oath (kaffarat yamin). If he is uncertain about the number of vows, he must err on the side of caution by offering expiation in a way that clears his conscience. As for making up missed prayers, he must repent to Allah and seek forgiveness, and make up the prayers he missed according to his ability. If he does not know the number, he should continue making them up until he is certain his conscience is clear, while observing the correct order between the missed prayers.