So, I'm starting to get somewhere. In order to create a new material, I had to remove all the usemtl lines from the OBJ file. Maybe there's some other way, but I couldn't figure it out. Also, hiding bits like lashes, eyeballs and brow overlays in edit mode did not hide it in texture paint mode, so I went ahead and removed those faces. I figure whatever modification of the original geometry lets me paint the texture in the way I want is fair game for now. Still, if there's an easy way to hide things in texture paint mode, let me know.

My first paint results obviously don't look like much, but that's just a matter of practice. I might have switch from mouse to tablet at some point to get a bit more control.

-- I'm not mad at you, just Westphalian.