I don't understand the question. Is the point that you are not insisting that the action of $G_x$ on a neighbourhood of $x$ be locally linear? If the action is locally linear, then surely $M/G$ is an orbifold by definition and the orbifold universal covering of $M/G$ is the ordinary universal covering of $M$.

But the meridian of the torus is contractible in the solid torus and represents a generator for the fundamental group of the Mobius strip.

Concerning the question at the end of your post: a wedge of $n$-manifolds will have top cohomology group a direct sum of $\mathbb{Z}$'s and $\mathbb{Z}/2$'s, so a mod-$p$ Moore space for $p$ an odd prime won't be of the form that you suggest.

I would change the definition to avoid having to view the void complex as a simplicial complex. The $-1$-sphere on the other hand is a perfectly decent space, defined as the points in $\mathbb{R}^0$ at distance 1 from the origin, a sort of boundary of the 0-disk.