What is the complexity class of solving hash decision problems?
It's in NP.  In particular, it is easy to provide a witness for a yes answer (the witness is such a value $p$) and given a witness, you can verify the yes answer in polynomial time.

As you say, it's not in P if the cryptographic hash function is cryptographically secure.

I don't think it's likely to be NP-complete, but at the moment I can't articulate a specific reason why I think that.

(If you have a one-way permutation, then it's in NP $\cap$ coNP, so unlikely to be NP-complete... but that doesn't apply to your question.)