What's the point in these theorems?

Sometimes math is weird. Often you know exactly why a particular math thing is interesting. Like it's so obvious that algebraic geometry is there to help figure out where really weird multinomials intersect. But other times, even for simple things for which there's lots of research and historical precedence for concern, I just don't get it. Here's a list of things I just don't get. I don't understand the point of pursuing them. I understand the mathematical process, I just don't get the point:

• Craig's Interpolation Theorem in logic, if a implies c, then there exists b such that a implies b and b implies c and b only involves the intersection of vars from a and c
• Curry's paradox and Löb's theorem - I have trouble following the elementary proofs of these. They seem to say you can prove anything "'if X is the case then Santa Claus exists' proves Santa Claus exists' or something
• Herbrand's theorem - proves universals using examples?
• the Deduction theorem - it just seems so obvious. It's just Modus Ponens, right?
• quadratic reciprocity - allows computation of square roots in modulo arithmetic. Why you would want to do that, I don't know

I want to understand these things, and I can (usually) follow step by step manipulations, but I just don't get what they are for and what the point is.