I'm pretty sure it's her Void which allows his movements. As I said before, when Shu activates Inori's Void, he undergoes a transformation. The most obvious fact is that his eyes change. It seems he also gains super-speed, super-jump, and of course, super-strength. I don't know why her Void is so special, but I'm placing my bets on the theory that Inori is somehow connected to Shu from the past. Maybe Inori and Shu were the cause of Lost Christmas (and subsequently the creation/development of Voids). A long stretch, I know... but I haven't read any other theories yet.

As far as fighting "with" instead of "without"... from what we currently know about Voids, it won't happen. It's almost like pulling the soul from the body, thus why there are no memories, and they lose consciousness. However, based on my theory again, if Inori is somehow connected to the creation of Voids, then maybe she knows, or will find a way to allow her void to be used while conscious, thus being able to fight alongside Shu.