Are all mathematical problems solvable?

Nope. Not if physics is computable.

Are they all solvable to humans?

Nope. Not if human minds are computable.

Not all true theorems have a proof? (what does that even mean)

It means exactly that your Turing machine enumerating all possible texts may never halt. What does it mean in terms of the validity of the theorem? Nothing. The truth value of that theorem may be forever inaccessible to us without appeal to a more powerful axiomatic system or without access to a hypercomputer.