Historically, reverse math is very closely tied to ideas in proof theory, but as Andreas points out, over the last decade or so, the connection to recursion theory has been very strong. For the foundational ideas in the area (in Simpson's book, for instance, or many of Friedman's writings) proof theory is very relevant.
Probably the two main introductions to proof theory right now are:
- Proofs and Types by Girard (available online)
- Basic Proof Theory, by Troelstra and Schwichtenberg
(A note on the Troelstra and Schwichtenberg book: the book is harder than most textbooks to get through solo, because it's very detailed; looking through it solo requires more than the usual amount of work identifying for oneself what the big picture is.)
Finally, in regards to your question about whether a statement has only finitely many proofs: proof theory is definitely the right place to look for questions like that. As Andreas points out, it turns out that it's very hard to phrase questions like that coherently, because it's easy to modify a proof in a trivial way, but hard to define precisely what should constitute a trivial modification.