Can I quantify the word "proof", so is it possible to write the phrase many proofs and if yes in which situation is this more appropriate than writing much proof?
Yes, proof can be quantifiable. One obvious context is the realm of mathematics, where there may be more than one proof of a theorem. So, for example, one could say:
There aren't many proofs for the four-colored map problem.
Many early proofs of Fermat's last theorem were found to contain errors.
On the other hand, proof is sometimes used in a way similar to evidence. (One might argue that evidence would be a better word to use in such situations, but the fact remains that proof is nevertheless used.) For example, in this science editorial, the author wrote:
There is much proof that Mars was once, if not a green planet, a planet that once held flowing water and possibly life.
Both can be used; it really depends on if you're talking about a body of evidence or individual proofs.
As a footnote, other usages of the word proof are clearly quantifiable, such as:
The photographer gave us nine proofs to choose from.