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As far as I know, the computer proofs run an enormous amount of iterations of the problem, perhaps regressively, until they reach a solution or contradiction.
There was contention over whether the computer proofs were mathematically sound or not.
I don’t have the expertise to weigh in on that controversy.
I think it was settled that the computer proofs are acceptable to mathematicians.
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Yes, fermat is a special case of the Langlands program.
No, wiles' proof did not require computers other than what is fairly common for ultra complicated theoretical proofs: for verification purposes.
A famous proof that was achieved with computers is the map colouring one.