So we just keep fighting the battle as if my EC was his? And if I win the battle does my EC get discarded or returned to my territory?
He has control of the EC because he banded to it. He can play matching enhancements on it as usual. If the EC survives, it returns to your territory. Otherwise, whatever happens to it (such as discard, capture, convert, etc.) happens to it as normal.
Also, can set aside enhancements be played in my prep phase and discard phase?
You can use a set aside that targets your own characters during prep and discard phases. For example, you could play Two Possessed by Demons during your prep phase to set aside your own humans but not your opponent's humans.