Concerning the proof that in Riemannian setting proper discontinuity is equivalent to discreteness, even more is true: If $G$ is a subgroup of group of isometries of a Riemannian manifold $M$ so that (for some $x\in M$) $d(x,g_i(x))$ is bounded for a sequence $g_i\in G$, then the sequence $(g_i)$ contains a convergent subsequence (Arzela-Ascoli theorem). Thus, discreteness of $G$ implies proper discontinuity. For your question about $G$-structures: it depends on what you mean: For $(X,G)$-structures, the best reference I know is Thurston's book "Three-dimensional geometry and topology." if you mean $G$-structures as in "reduction of frame bundle to various subgroups $G\subset GL(n,R)$", then read Kobayashi-Nomizu or Kobayashi's "Transformation groups in differential geometry".