So just two questions so I understand correctly:

Sport mode is just a version of Automatic (either Sport or Drive)? If you are driving in Manual mode, shifting gears manually, Sport (or Drive) mode doesn't apply, correct?

Also, what is this "M Mode" that others have referred to. Is it just a button/menu that can give a combination of sportier suspension, sportier dynamic stability, more linear gas pedal mapping, etc.?

I'll try to answer what I know.

The transmission is an automatic, has a torque converter not a clutch or computer controlled clutch system like the SMG or DCT. When you are in sport mode, the tranny won't change gears until at least 3000 rpm and holds gears to rev higher when you are in turns and driving aggressive etc. Also, the shifts are much faster and crisp compared to drive. It will also downshift when you are slowing down so it's in a more optimal gear to then just take off again, should you need/want. If you are in drive, you can shift the gear manually via the paddle shifters. The shifts are kinda slow and it eventually goes back into automatic drive mode. When in Sport mode and you click the paddles, it will stay in true manual mode.

M mode is programmed via the COMAND. There's several settings, one of which being the engine throttle sensitivity control. When in sport or M mode, the accelerator is MUCH more sensitive. For example, if you push the accelerator 50% down, you will get like 75% of the power. At WOT, it's the same in either M mode or non M. In addition, it opens up an exhaust baffle to make it louder and you can program the EDC suspension control to switch to sport setting (which also tightens up the variable steering wheel), should you choose. Oh and you can choose the level of the DSC as well. Either leave DSC on all the way or put it in MDM (m dynamic mode) which allows a little bit of oversteer.

I think that about covers it. Hope this helps.