ELT courses need pedagogical and content knowledge; the knowledge of professional teaching is as important as content or subject matter knowledge. you should enhance them
As a literature student one often might takes courses on teaching methodology as well. Furthermore, if you have the passion for teaching, I don't see why can't you teach ELT classes.
It depends on institutional policies or national policies, but if no restrictions or policies on that matter exist, then, of course any one with literature degree can teach ELT courses.
Well, I’ve majored in both English language and corresponding literature and recently I’ve finished my master’s in literary theory, I also intend to take a doctor’s in literary theory and criticism again, but I still work with ELT. I’m always studying both and take courses on the field in a regular basis so I don’t see why not.