This is going to sound a little odd, but I had something similar due to my second hard disk not working correctly. The player would load the playlist correctly, but would then fail to load the track from the second disk. It would then skip tracks. I think that if the player skips too many tracks, it pauses playback. It usually happened at a track transition, rather than in the middle of the track, but you never know...

In my case, it was either a loose IDE cable, or a power problem (I reseated the cable, and I had the car battery replaced at a service a few days later).

This might not be your problem, but you're going to take forever trying to track down phantom button presses (although Hijack has some nifty stuff that can help you), so rule out HD and power supply problems first. The fact that it works fine on AC suggests to me that it might be a power supply problem.