I can give you a partial solution for the commuting with 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 $P$.