For the first question, if the map of the surface in the manifold is made generic, then the singular set will consist of just double and triple points, and will be a graph with vertices of valence six (and maybe closed loops). Near the triple points, it will look like:

As for the second question, the complement of the self-intersection locus will just be subsurfaces, and can change quite a bit under homotopy. If the surface is made minimal area (in its homotopy class), then some information can be gleaned from a paper of Freedman-Hass-Scott. Minimal area maps of $\pi_1-$injective surfaces are immersions, and may factor through a finite covering. They show that the minimal immersions intersect minimally in a certain sense. From their results, one can show that the complementary regions of the singular locus will be $\pi_1$-injective (of course, this is meaningless if all of the complementary regions are disks). I think one could perturb this immersion to be generic while retaining this property. Not sure what exactly you're fishing for though.