What is the big-O (worst-case upper bound) for time and space requirement of the different Chomsky classes?
Regular languages can be accepted in linear time and constant space.

Valiant's algorithm parses arbitrary context-free languages in time $O(n^\omega)$, where $\omega$ is the matrix multiplication constant. Deterministic context-free languages (which are those used in practice) can be parsed in linear time.

Context-sensitive languages are identical to the class $\mathsf{NSPACE}(n)$. I'm not aware of any non-trivial time upper bound.

Recursively enumerable languages cannot be parsed in general, since $\mathsf{R} \neq \mathsf{RE}$.

The $\mathsf{P}\stackrel?=\mathsf{NP}$ question is completely unrelated. Unfortunately, the undergraduate curriculum usually spends much more time on the old-fashioned Chomsky hierarchy compared to more modern hierarchies afforded by computational complexity theory.