I don't recall Wiles' proof assuming that p >= 11 – can you give a reference? I can't find one quickly.

I think this is in the original paper that modularity implies FLT, but I'm on vacation and don't have a copy available to check. Does this suffice as a reference?

Ben Green, not Greenberg

Yes, thank you.

They were subsumed by Kummer's work, which I understand to have been motivated more by a desire to understand algebraic number fields and reciprocity laws than by Fermat's last theorem in particular. For this, he developed the theory of ideal numbers, which is very general.

Sure, but Kummer was aware of the literature before him, and almost certainly used their results to guide him.

Sure, but the ultimate significance of the work remains to be seen. Of course, tastes vary, and there's an element of subjectivity, but I think that we can agree that even if there's a case for the proof being something that people will find interesting in 50 years, that the prior in favor of it is much weaker than the prior in favor of this being the case of, e.g. the Gross-Zagier formula.

Agreement may there depend very strongly on how you unpack "much weaker" but I'd be inclined to agree at least weaker without the much.