Classes between $\textbf{PSPACE}$ and $\textbf{EXP}$ 1) What classes contain $\textbf{PSPACE}$, are contained in $\textbf{EXP}$ and (presumably) are not equal to $\textbf{PSPACE}$ nor to $\textbf{EXP}$?

A possible class satisfying this requirement: the class of languages that can be recognized in exponential time and subexponential memory.

2) Which of these classes are provably strictly larger than $\textbf{P}$?
Depends what you mean by "class". 

I don't know of any natural class that satisfies that, but certainly one can do the following:

Consider the set of languages that are decidable in time $2^n$ (exactly, not just any exponential). This is strictly contained in EXP by the time-hierarchy theorem.
Take the union of this set and PSPACE, and you have a class that strictly contains P (again, time hierarchy), contains PSPACE, and is contained in EXP (and presumably, if PSPACE$\neq$ EXP, then there is also strict containemnt here).