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 in the case of a reductive group over a local non archimedean field, but it seems to me that you will study only very few representation with this (finitely many?).
Edit: The observation with cuspidal mapsto supercuspidal and parabolic induced to parabolic induced is best explained with the restriction induction formula for composing restriction and induction, perhaps modulo the assumption that supercuspidal are induced from maximal compacts (I do not know, if this is true in the generality you want this.)