If P = NP, can all NP problems be solved within time $O(n^k)$ for fixed $k$?
Yes, the answer the the question is "false".

By contradiction, assume every problem in $NP$ has a $O(n^k)$ algorithm, for some fixed $k$. That is, $NP \subseteq {\sf DTIME}(O(n^k))$

This would contradict the time hierarchy theorem, which implies the strict inclusion

$$
{\sf DTIME}(O(n^k)) \subset
{\sf DTIME}(O(n^{2k}))
$$

However, we also have

$$
{\sf DTIME}(O(n^{2k})) \subseteq NP \subseteq
{\sf DTIME}(O(n^{k}))
$$

which contradicts the previous strict inclusion.