Regarding question 1: I am pretty sure that only the full proof has given us FLT for infinitely many prime exponents. As soon as we know that there are infinitely many regular primes, we will have an alternative route to FLT for infinitely many prime exponents, but I am going to stick my neck out and say that that's unlikely to be easy. So at present, the answer is: to prove FLT for infinitely many exponents, you need the modularity theorem (at least for semistable elliptic curves).

Regarding question 1: I am pretty sure that only the full proof has given us FLT for infinitely many prime exponents. As soon as we know that there are infinitely many regular primes, we will have an alternative route to FLT for infinitely many prime exponents, but I am going to stick my neck out and say that that's unlikely to be easy. So at present, the answer is: to prove FLT for infinitely many exponents, you need the modularity theorem (at least for semistable elliptic curves).

Edit: to be clear, Kummer's proof that FLT holds for regular prime exponents is of course much simpler than the proof of the modularity theorem. Moreover, there is a probabilistic heuristic that suggests that more than half of the primes are in fact regular, and this agrees extremely well with numerical data. The "unlikely to be easy" comment referred to the missing tile of the puzzle: namely to turn the heuristic into a rigorous proof for infinitude of regular primes.