I think the answer to the first question is yes and the answer to the second one is no:
Yes, the quotient is an orbifold. The action of the finite group $G_x$ in a neighbourhood of $x$ can be linearized (at least if the action is by diffeomorphisms, I don't know about $C^0$ regularity), and the quotient $M/G$ is locally modelled on $G_x \backslash T_xM / T_x (G\cdot x)$.
No this orbifold is not good in general. For instance, you can glue a solid torus with a trivial circle fibration to a solid torus with a Seifert fibration with one singular fiber in the center and get a closed Seifert 3-manifold with one singular fiber. The fibration is given by the orbits of an action of S^1 and the quotient orbifold is a sphere with a single orbifold point, the simplest example of an orbifold not covered by a manifold.
More generally, I think every orbifold $M$ if dimension $n$ is the quotient of a manifold $P$ by an almost free action of the orthogonal group $O_n$ ($P$ is the principal $O_n$-bundle associated with the orbifold tangent bundle equipped with an orbifold Riemannian metric).