Why AKS primality test is called polynomial?
The time complexity of the AKS primality test is $O(n^6)$, where $n$ is the length of the input. The length of an integer $N$ is roughly $\log N$, so the time complexity of the AKS primality test applied to the integer $N$ is $O(\log^6 N)$.

Finally, I'm not sure what you mean by a problem being strictly contained in P. Every problem in P is contained in some smaller class $\mathsf{TIME}(O(n^k))$ for some $k$.