I can give only an answer to the 2nd question:

The functor $\mathcal{F}$ is a composition of restriction to $G(o)$ and then projection onto the subrepresentation of $G(o)$, which have kernel $\bmod p$.

Both of these functors are very wellbehaved by the Peter Weyl Theorem.

So all the properties you want follow, but it seems to me that you will study only very few representation with this (finitely many?).