First let's consider constructible abelian sheaves for the etale topology. Higher direct images under *proper* morphisms carry constructible abelian sheaves to constructible abelian sheaves. The point is that for *proper* curves over separably closed fields the finiteness and cohomological vanishing beyond dimension 2 hold even for $p$-torsion in characteristic $p>0$ since one can use the Artin-Schreier sequence in place of Kummer theory to make contact with coherent cohomology (this is explained in Milne's book on etale cohomology, for example). Consequently, the usual fibration method to prove preservation of constructibility for torsion orders invertible on the base also works in general (i.e., without hypotheses on torsion orders) in the proper case.

The proof that a (constructible) $\mathbf{Z}_{\ell}$-sheaf becomes lisse upon restriction to members of a stratification is rather soft and so works without any assumption on $\ell$, and by the preceding the usual proof (such as 12.15 in Chapter I of the book by Freitag and Kiehl on etale cohomology) that ${\rm{R}}^if_{!}$ carries constructible $\mathbf{Z}_{\ell}$-sheaves to constructible $\mathbf{Z}_{\ell}$-sheaves works for *proper* $f$ without any hypothesis on $\ell$. Question (1) (for which I assume you meant for $U$ to be *dense* in $Y$, so not empty for example) therefore has an affirmative answer for ${\rm{R}}^i f_{\ast}(\mathbf{Z}_p)$ for any proper map $f:X \rightarrow Y$ between noetherian schemes and any prime $p$, so tautologically also with $\mathbf{Q}_p$ in place of $\mathbf{Z}_p$.

Question (2) has a rather negative answer for any meaningful notion of "almost trivial"; even for a non-isotrivial pencil of ordinary elliptic curves the resulting representation into $\mathbf{Z}_p^{\times}$ typically has open image. So without a more precise meaning assigned to "almost trivial", one can't really say anything more (and it isn't explained why question (2) would be expected to possibly have an affirmative answer).