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. There is a natural map $\phi_n : \Theta \to \Theta$ induced by $z \mapsto z^n$. Given an effective Cartier divisor $D \subset X$, there is an induced morphism $X \to \Theta$. Then the $n^{th}$ root stack can be given as the following fiber product.
$\require{AMScd}$
\begin{CD}
\sqrt[n]{D} @>>> \Theta\\
@V V V @VV \phi_n V\\
X @>f_D>> \Theta
\end{CD}
In particular, takng $(X,D) = (\mathbb{A}^1, 0)$, $f_D$ is the standard smooth cover of $\Theta$ by a $\mathbb{A}^1$ and $\sqrt[n]{D}$ is Deligne-Mumford with finite inertia and proper over $\mathbb{A}^1$. Thus $\phi_n$ is a proper tame morphism with finite inertia and which is representable by Deligne-Mumford stacks.
By Theorem 3.1 in Abramovich, Dan; Olsson, Martin; Vistoli, Angelo, Twisted stable maps to tame Artin stacks, there exists a factorization of $\phi_n$ through a relative coarse moduli space
$$
\Theta \xrightarrow{\pi} \bar{\Theta} \xrightarrow{\bar{\phi}_n} \Theta.
$$
On the other hand, $\bar{\phi}_n$ is a representable proper birational morphism to a smooth stack so $\bar{\phi}_n$ is an isomorphism by Zariski's Main Theorem.
The formation of the relative moduli space commutes with arbitrary base change for tame morphisms. Thus the relative coarse moduli space of $\sqrt[n]{D} \to X$ is canonically isomorphic to $X$. By pulling back to a smooth a presentation of $X$ and working through the construction of coarse moduli spaces, we conclude that if $X$ has a coarse moduli space, so does $\sqrt[n]{D}$ and they are isomorphic.
Finally, $\sqrt[n]{D} \to X$ is representable by Deligne-Mumford stacks since the property of being representable by Deligne-Mumford stacks is compatible with base change. So if $X$ is Deligne-Mumford, we conclude that $\sqrt[n]{D}$ is as well.
