Pressing the "Start" button will tag in your other fighter. They will jump in from the same side that your current fighter is on.
Holding "Left" or "Right" and pressing "Start" will dictate which side of the screen they jump out from.
Hell yes they do!
The clock counts down slower than it does in normal 1v1 mode. The clock is reset to "99" whenever each character dies. 2v2 matches can last a long time (especially with the tug of war cheat).
There are limitations to how many times certain moves can be used in a kombo. Experiment!
There might be a cheat for that ;)