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Birational morphisms from DM stacks to their coarse moduli spaces

Yes. For each nontrivial element $g\in G$, the fixed points form a closed set, which must not contain the whole space as then $g$ would act trivially (by reducedness). The complementary open set thus ...
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One way to construct the root stack is to consider the universal situation of $\Theta := [\mathbb{A}^1/\mathbb{G}_m]$. Here let us work over $\mathbb{Z}[\frac{1}{n}]$ since we assume $n$ is invertible....