Another definition of local coefficients True or false: Let $A$ be a $\mathbb{Z}[\pi_1(X)]$-module. The homology of $X$ with local coefficients in $A$ is equal to the $H_*(\tilde{X} \times_{\pi_1(X)} A,\mathbb{Z})$. Why i'm interested: This would simplify the definition of homology with local coefficients considerably. Homology with local coefficients would then just be homology of a space called a 'local coefficient system'.

(I think, I do not understand your question in its entirety, because I don't know how to take a pushout of objects of two _different_ categories)

If you want to compute $H_*(X,A)$, you actually should take the complex $C_*(X,A):=C_*(\tilde{X},\mathbb Z) \times_{\pi_1(X)} A$.

In general, for an arbitrary $\pi_1(X)$-module the homology groups $H_*(X,A)$ do not equal the homology of some topological space, because $H_0(X,A)$ does not necessary equal $\mathbb Z$ even if $X$ is path-connected.