Is there a nontrivial class $S$ of smooth DeligneMumford stacks over a base $B$ with the property that if $\mathcal{X}, \mathcal{Y} \in S$ have isomorphic coarse moduli spaces (assumed to exist) then $\mathcal{X} \cong \mathcal{Y}$? If $B = Spec(k)$, $k$ a field (of characteristic $0$ if necessary), can one take $S$ to be the class of all irreducible, smooth, separated DM stacks with trivial inertia in codimension $\leq 1$?

Yes, the class of all smooth, separated DM stacks over a field of characteristic $0$, with trivial inertia in codimension at most $1$, over a field of characteristic $0$, has the propery you want. The point is that every moduli space of such a stack has quotient singularities; and every variety with quotient singularities is the moduli space of a unique such stack. I believe that this was first proved in Angelo Vistoli: Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97 (1989), no. 3, 613670, Proposition 2.8 (uniqueness is not stated there, but it follows from the proof). 


The answer to the second question is No. For example the weighted projective stack $\mathbb{P}(1,2)$ and $\mathbb{P}^1$ have isomorphic coarse moduli spaces but are not isomorphic stacks. More generally, you can always ``add stack structure" in codimension one by using the root stack construction first developed by Cadman. 

