If $\mathcal{X}$ is a normal Deligne-Mumford stack then its coarse moduli space $X$ is normal. Since $\mathcal{X}$ its normal its admits an étale atlas $U_i\rightarrow\mathcal{X}$, with $U_i$ normal schemes. Now, the statement follows because $U_i$ is normal and $G$ is a finite group acting on $U_i$ then $U_i/G$ is normal as well. In dimension one this translate into the fact that a quotient of a smooth curve is smooth.
In particular, if $\mathcal{X}$ is a smooth Deligne-Mumford stack the coarse moduli space $X$ has finite quotient singularities, that is étale locally it is isomorphic to a quotient of a smooth scheme by a finite group, and $X$ is normal.
Logarithmic geometry fits well for instance with moduli spaces. Indeed $\overline{M}_{g,n}$ is a moduli space for log curves. See
Fumiharu Kato, "Log smooth deformation and moduli of log smooth curves",
Internat. J. Math., 11(2), 215–232, 2000.
If $X$ is a smooth scheme, and $D\subset X$ is a divisor. We may define a log structure on $X$ with respect to the divisor $D$ as
$$M(U) = \{f\in \mathcal{O}_X(U) \; | \; f_{|U\setminus D}\in\mathcal{O}_{X}^{*}(U\setminus D)\}.$$
When $D$ is normal crossing it is log smooth. The answer to $(2)$ is negative. For instance consider $\overline{M}_{1,2}$. From the analysis of the singularities of $\overline{M}_{1,2}$ below you can see that $\mathcal{E}xt^1(\Omega_{\overline{M}_{1,2}},\mathcal{O}_{\overline{M}_{1,2}})$ is not trivial. Then $\overline{M}_{1,2}$ is not $d$-semistable.
As I wrote if $\mathcal{X}$ is a smooth Deligne-Mumford stack the coarse moduli space $X$ has finite quotient singularities. A nice example is the moduli space of pointed curves $\overline{M}_{g,n}$. If $g = 0$ the stack coincides with the coarse moduli space because $n$-pointed rational stable curves are automorphism-free. Anyway one can see the difference between the stack and the coarse moduli space already for $g = 1$.
It is well known that $\overline{M}_{1,1}\cong \mathbb{P}^{1}$ and $\overline{\mathcal{M}}_{1,1}\cong \mathbb{P}(4,6)$. Clearly $\mathbb{P}^{1}\cong \mathbb{P}(4,6)$ as varieties. However they are not isomorphic as stacks. Indeed $\mathbb{P}(4,6)$ has two stacky points with stabilizers $\mathbb{Z}_{4}$ and $\mathbb{Z}_{6}$. These two points are indistinguishable from any other point on the coarse moduli space $\overline{M}_{1,1}$.
For singularities we have to look at the case $g=1, n=2$. Since $\overline{\mathcal{M}}_{1,2}$ is a smooth Deligne-Mumford stack the coarse moduli space $\overline{M}_{1,2}$ will have finite quotient singularities at the places where the automorphism groups jump.
Indeed $\overline{M}_{1,2}$ is a rational surface with four singular points. Two singular points lie in $M_{1,2}$, and are:
- a singularity of type $\frac{1}{4}(2,3)$ representing an elliptic curve with automorphism group $\mathbb{Z}_4$ and marked points $[0:1:0]$ and $[0:0:1]$;
- a singularity of type $\frac{1}{3}(2,4)$ representing an elliptic curve with automorphism group $\mathbb{Z}_6$, and marked points $[0:1:0]$ and $[0:1:1]$.
The remaining two singular points lie on the boundary divisor, and are:
- a singularity of type $\frac{1}{6}(2,4)$ representing a reducible curve whose irreducible components are an elliptic curve with automorphism group $\mathbb{Z}_6$ and a smooth rational curve connected by a node;
- a singularity of type $\frac{1}{4}(2,6)$ representing a reducible curve whose irreducible components are an elliptic curve with automorphism group $\mathbb{Z}_4$ and a smooth rational curve connected by a node.
For other manifestations of these phenomena you may look at stacks of weighted pointed curves, prym, spin, and level curves (http://arxiv.org/abs/1205.0201).
Remark: Note that a singular stack could have a smooth coarse moduli space. For instance, take $X = L\cup R\subset\mathbb{A}^2$ be the union of two lines. We have an action of $S_2$ on $X$ switching $L$ and $R$. Now, the quotient stack is singular because it admits an étale cover by something singular. However, the coarse moduli space is $\mathbb{A}^1$.
