I can give only apartial answer to the 2nd question, but it will be too long for a comment:

For the parabolic induction: 

The Iwasawa decomposition gives you

$$ Res_{G(o)} Ind_{P(F)}^{G(F)} \chi \cong Ind_{P(o)}^{G(O)} Res_{P(o)} \chi $$

Then you find that

$$ \left( Ind_{P(o)}^{G(o)} Res_{P(o)} \chi \right)^{\Gamma(p)} =  Ind_{P(f)}^{G(f)}  \tilde{\chi} $$

where $f$ is the residue field. Where $\tilde{\chi}$ is the irreducible sub representation of $\chi$ by the $M(p)$ invariant vectors for the Levi subgroup $M$ of $P$. 

For the observation with supercuspidal, I have only a comment:

I assume that the super cuspidal is induced from an open compact group $G(o)$, then this is essentially just Frobenius reciprocity

Consider a representation of $\chi \in G(o/p)$

$$ Hom_{G(o)}(\chi, Res_{G(o)} Ind_{G(o)}^G \pi) = Hom_G (Ind_{G(o)}^G \chi, Ind_{G(o)}^G \pi)$$

for $GL(n)$, there you will have to see for which representation the induced one is actually irreducible and supercuspidal (see e.g. Kutzko and Bushnell)

For exactness:

Restriction to open compact subgroup is always an exact functor, since the representation theory of compact groups is completely irreducible.