Should I (can I?) temporarily unmount breaker box to drywall behind?
It's probably also secured by nails behind the panel. 

Actually, you don't want to do that.  That plywood has an important utility purpose, it's where the power, phone and other utilities attach.  You need it.  

The one thing I would do is extend the plywood to span out to the joists.   Feel free to lift up the cables and paint under them, might be hard to tell it's not drywall.  Don't paint the cables, it's important to be able to read their markings.  It's utility area, it's supposed to look ugly.