We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group of deck transformations.
Now, I would like to understand this:
1.) Does the group of deck transformations to a given covering map always act properly and discontinuously on the covering space?
2.) If I have a covering map $p:X \rightarrow Y$ and I look at the group of deck transforms $G(X,p)$. Does it then follow that $Y$ is isomorphic to $X/G(x,p)$?
For (1), if $G$ is assumed to be a group of homeomorphisms of $X$, then the fact that $\pi \colon X \to X/G$ is a covering map is equivalent to $G$ acting on $X$ freely and properly discontinuously (this is Thm 81.5 in Munkres).
For (2), there is indeed a homeomorphism $X/G \simeq Y$ if we assume that the covering map $p \colon X \to Y$ is regular (that is, the action of $G$ on the fibers is transitive).