Is a quotient space of a smooth manifold by a properly discontinuous group action also able to be smooth manifold?

Question

$X$ is a smooth manifold (Hausdorff, locally Euclid) with an action by a group $G$. I have the following questions.

1. Quotient space $X/G$ has a quotient topology. Does $X/G$ have Hausdorff property?
2. Let $\pi : X \to X/G$ be a quotient map. Is $\pi$ locally homeomorphic?