# Do poly-time algorithms exist whose time complexity is unprovable?

If not, is there a decision procedure that successfully classifies any polynomial time algorithm as poly-time within a time polynomially bounded by the length of the input algorithm?

On input $x$, go over all strings of length $\log |x|$, and check if one of them is a proof of the consistency of A. If so, loop for $2^{|x|}$ iterations. Otherwise, halt.
Your second question asks whether there is an algorithm that, given a Turing machine $T$, answers in polynomial time whether $T$ is a polynomial time Turing machine. Unfortunately, this is undecidable. Suppose that we only consider Turing machines which don't depend on their input (say the first thing they do is erase their input). Such a machine $T$ runs is polynomial time if and only if it halts on the empty tape. Deciding whether a Turing machine halts is impossible.