I have a question about the translation groupoid mentioned in the last paragraph. I don't understand why $N_x/G_x$ is an open embedding.
I think because $X$ is the orbit space, we need to mod out by those elements in $G_1$ that map elements in $N_x$ to $N_x$. It seems to me that those elements are just the union of $O_g$ where $g \in G_x$, whereas $G_x$ is just a subset of this union.
If we have $h \in G_1$ such that $s(h),t(h) \in N_x$, then, indeed, $h \in O_g \subseteq W_g$ for some $g \in G_x$. Furthermore, by definition, we have $g \cdot s(h) = \tilde{g}(s(h)) = t(h)$, so that the action of $G_x$ on $N_x$ acts just as wanted: it identifies objects $a,b \in N_x$ precisely if there is an isomorphism between them.