ag.algebraic geometry - coarse moduli space of DM stacks

No, not every DM-stack has a coarse moduli space. The following is a counter-example (see my paper on geometric quotients):

Let X be two copies of the affine plane glued outside the y-axis (a non-separated scheme). Let G=Z₂ act on X by y → –y and by switching the two copies. Then G acts non-freely on the locally closed subset {y=0, x ≠ 0}. The quotient [X/G] is a DM-stack with non-finite inertia and it can be shown that there is no coarse moduli space (neither categorical nor topological) in the category of algebraic spaces.