1. By the [Hochster-Roberts Theorem][1] it is Cohen-Macaulay
 2. The canonical sheaf of $\mathbb A^n$ is $G$-invariant (because the elements of $G$ have det=1) and hence it descends to the canonical sheaf of the quotient, which is then a line bundle.

  [1]: https://en.wikipedia.org/wiki/Hochster%E2%80%93Roberts_theorem