I mount mine on the left but you could mount it on the right, although I don't think it would be very comfortable that way if you were holding the camera at all. This would also put the shutter release on the bottom instead of the top for shooting verticals.
Since the mounting screw slides along the bottom, you can mount it out with enough room attach video/flash/usb/release cords. I haven't tried it this way, but I suppose as long as you tightened it enough you wouldn't have a twist problem.
Speaking of tightening it down, another benefit of this bracket over others is that the mounting screw works with both an allen wrench and a coin/flathead-- a really nice design which competitors should offer for when you inevitably lose the wrench.
I don't find changing batteries to be an issue when using this with the Mark III because they last so long!
Also, I believe that the universal RRS L plate is only for smaller bodies, without a vertical grip.