It is Canon's LP-E6 battery and is rechargeable. For shooting photos, it'll last something along the lines of 300-500 shots depending on how you shoot. I don't shoot video, so I couldn't tell you how long they will hold out for video.
Yes, it is. I often do this when I'm shooting a scene with a broad dynamic range of light and shadow that would be impossible for the camera to capture in one exposure. So I go to AEB (automatic exposure bracketing) and set it to take one exposure set for the highlights (+2), one exposure set for the mid tones (0), and one exposure set for the shadows (-2). I have my camera set on a tripod, set for aperture priority (Av), and set for one shot and the self-timer set for 2 secs. The camera, when you push the shutter once, will then, after the 2 second delay, fire three shots in succession, giving you three exposures that can be combined into one HDR (high dynamic range) image. I have software that combines these in my computer, but the Canon EOS 5D Mark III can now also do this combining of the three exposures in-camera. I hope this helps answer your question.
Yes, its a L lens, the 24-105. It would be about $800 by itself. I a incredibly useful lens, F4 is fine for almost any situations except very, very low light. This lens covers photo and video very well.
No. I've used both and if you already got the M2, I don't think it's worth the expense. Image quality up to ISO 1600 is very similar. The auto focus is better on the M3 but it's not worth the time and money you would be spending. Video quality is basically identical.