This must have been known to the ancients, but I am having some trouble finding the references: what can be said (especially geometrically) about the normal closure of an element in a surface group? Especially an elliptic element (in an orbifold group)...

This doesn't say much about the elliptic case, but there is Hempel's theorem that the normal closure of an element $\alpha$ contains a (power of a) simple curve $\beta$ if and only $\alpha$ is a (power of a) simple curve. Furthermore, any such $\beta$ is either a power of $\alpha$ or a commutator of $\alpha$ with a simple loop $\gamma$ intersecting $\alpha$ in one point. You should be able to glean some structural information about the normal closure from his paper. John Hempel, Onerelator surface groups, Mathematical Proceedings of the Cambridge Philosophical Society (1990), 108: 467474 

