>>1 What should be in a proof is something not even all mathematicians agree on. Constructivists, in particular, only allow proofs with things that can be constructed- no infinities, no excluded middle, no proofs by contradiction.
At first Constructivism was seen as a threat by a lot of non-constructivist mathemeticians, but then even non-constructivist began to think that even if you are OK with a proof that says "There is a number that ..." it is nice to have another proof of the form "Here is a number that ..."
The Mathematical Experience by Philip J. Davis and Reuben Hersh covers this and a lot of other philosophical/historical issues on math.