Are you aware of the work of Bob Oliver, in particular his article `Fixed-Point Sets of Group Actions of Finite Acyclic Complexes', Comment. Math. Helvetici Volume 50 (1975) 155-177? He classifies the finite groups that can act without a fixed point on a finite acyclic complex. The situation is quite different from the 2-dimensional case that you mention in your question. His article does not consider universal spaces with respect to any family, but the techniques he uses should be relevant.

Martin Bridson's formulation of the triviality question on Bestvina's problem list is a nice one, but the question already appeared in the 1965 version of the Kourovka Notebook as question 1.12 (asked by Greendlinger who attributed it to Magnus).

I imagine that you are aware of the Lyndon-Hochschild-Serre spectral sequence, with $E_2^{i,j}= H^i(P/Z)\otimes H^j(Z)$ and converging to a filtration of $H^*(P)$? As Denis T says, the answer to your question is probably `no' in general, but not many computations have been done.

I am unsure whether this is correct, so I didn't want to put it as an answer. However, I think that the functor that sends a simplicial set to the simplicial abelian group whose $n$-simplices are the free abelian group with basis the $n$-simplices of the original simplicial set does this job.

You need to add that $\gamma$ has infinite order.

At risk of stating something that you know already, but the natural correspondence is between covering spaces and subgroups: if $X$ is an aspherical 2-complex with $\pi_1(X)=G$ then there is a covering space $Y$ of $X$ which is also aspherical and 2-dimensional and has $\pi_1(Y)=H$.

Maybe Stefan's question about $\mathbb{Z}_p$ group schemes should be posed as a new question as he suggests.