Hey!
1. You can absolutely just make an attack with an enemy during the GM Turn.
2. Part of the game is that resource management, so I shy away from the idea of starting at being fully restored for each fight. In LIGHT, I write about the idea of including "Wells" which are recharging stations sprinkled throughout a mission, but they are limited in number and effectiveness. You can certainly increase the amount of recovery, but generally I think that takes away from some of the tension of maintaining those resources across a session.