Is the complement of MAX-CLIQUE in NP?
$MAX-CLIQUE \in \mathsf{coNP \Leftrightarrow NP = coNP}$.
The answer is yes: your variant of $MAX-CLIQUE$ is in $\mathsf{P^{NP}}$, since knowing the size of max clique you can solve your problem.


There is no polynomial verifier (if $\mathsf{NP \neq P^{NP}}$) for this problem, because it is not in $\mathsf{NP}$.

Classic $MAX-CLIQUE$ which only asks if there is a clique of size at least $k$ is $\mathsf{NP}$-complete. But if $\mathsf{P = NP}$ you can ask a machine if there is a clique of size $|V|, |V| - 1, |V|-2...$ until the answer is "YES" to find max clique. Of course, you can use binary search instead of linear. 

So, $\mathsf{P = NP}\Leftrightarrow MAX-CLIQUE\in \mathsf{P}$.