Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$.

* Question.  Is there a reference for such a statement? At least in dimension 3?

The reason why this should be true is that it should be possible to find such a resolution of $X$ that the preimage of each point in $\tilde X$ is simply connected ($\mathbb CP^n/G$ is simply-connected for finite $G$). Then everything follows from a standard topological lemma.
I guess this statement should be true as well if $X$ is a complex analytic variety with arbitrary Kawamata log terminal singularities (because Fano manifolds are simply connected). I would be grateful for a reference for any kind of such statement.