If I remember correctly, any proof of P = NP would be have to be constructive, so it would immediately give us a fast algorithm for ALL NP-complete problems, as they can all be reduced to each other.
It isn't actually _necessary_ for a P=NP proof to be constructive. A constructive proof would probably be easier to reason through, but it is prossible to prove the existence of an algorithm without providing one.
Not so. Pure existence proofs can be non-constructive.
The first historical example of an important proof which was non-constructive was Hilbert's Basis Theorem in 1888, which solved a famous problem posed by Paul Gordon. It is now viewed as in some sense being the first salvo in the debate over constructivism in the 20th century. See http://en.wikipedia.org/wiki/David_Hilbert#The_finiteness_th... for more.