Is the P vs NP problem, an NP or co-NP problem?
It isn't meaningful to say that a single specific question is in NP or any other complexity class. In order to classify things as being in P, or NP, or co-NP, etc. they need to be sets of problems with some parameter. So for example, the problem "Is the positive integer n prime" is a question where we can discuss this sort of thing. 

In a certain overly pedantic sense, the correct answer to your question is that the problem you care about is in P, since there's an algorithm which will answer every question of your form in polynomial (in fact constant time). But we don't know if that algorithm is just "Return  TRUE" or is "Return False." 

The idea that the video may be touching on is the idea that if P != NP, then in general, finding proofs is hard. In fact, "Is statement S provable in formal system A with a proof of length at most x" is an NP-hard problem" (Questions in this are things like "Is there a proof of Fermat's Last Theorem that's shorter than 5 pages.") . And we also expect that if P != NP, for most "natural" NP-complete problems, random instances will often not be easy. So if P != NP, then searching for proofs should often be difficult, and thus we shouldn't be surprised that the proof that P != NP would itself be one of those. This may be what they were trying to say.