To answer the question posed:

A proof is valuable because it helps convince oneself and others of the validity of that result from the axioms [whether those axioms are consistent or not]. Mathematicians, I believe, are not worried that the axioms are inconsistent, but rather hopeful that they are consistent; or even more precisely, optimistic that if the axioms are inconsistent they can be modified [if necessary] to be consistent and still encompass most things proved. But even if they are inconsistent, we won't figure that out without lots and lots of proofs in the meantime.

To answer the philosophical question from a personal point of view:

From my point of view, I do mathematics because I love certainty and truth. I also enjoy discovery. Godel's theorems simply tell me that there are some things I will never be certain of or discover (inside a formal system). This may be disappointing, but at some level we all have to deal with uncertainty. For example, I could be deceiving myself that I'm typing this message. But I (and most others I know) are willing to accept a few things on faith; and if shown we are wrong, modify our beliefs accordingly.