What you're asking about is substantially easier than 2.2.0.1 (it's easy to see that Hom spaces are compatible with the homotopy coherent nerve; the difficulty is to prove the analogous statement for the left adjoint). See kerodon.net/tag/01LA.

If $B\mathbf{Z}_p$ is of finite homotopy dimension then it also locally of finite homotopy dimension, because every finite index subgroup of $\mathbf{Z}_p$ is isomorphic to $\mathbf{Z}_p$. For an example of something locally but not globally of finite homotopy dimension: the slice $\infty$-category $\mathcal{S}_{/X}$, where $X$ is any space which is not finitely dominated.

Every $A$-module $M$ admits a canonical $\varphi_A$-semilinear endomorphism, given by $\varphi_M = 0$.

I don't think this conjecture can be true. Let $(A,I)$ be a perfect prism. Every free $A$-module $M$ of finite rank defines a prism $(A \oplus M, I \oplus IM)$. If you had such a functor, you could apply the "TP version" and quotient out $TP(A/I)$ to get a $TP(A/I)$-module $F(M)$, free of the same rank as $M$. Since $F$ is an additive functor this would need to come from a map associative ring spectra $A \rightarrow TP(A/I)$, which usually can't exist.