While I have the HTC Inspire, I am encountering the same issue with the music randomly starting.
I was playing a game with my headphones in and suddenly my music would start for no apparent reason. I would go in and hit the pause button and go back to my game. Then it would start up again a few minutes later while I was playing my game. Frustrated and thinking that it was the media player I was using (Meridian) I uninstalled it. Fixed, right? Wrong. After uninstalling it and going back to my game (Angry Birds), it redialed the last person I talked to on this phone. It did this several times while playing the game and I finally had to put my phone into airplane mode. It would still attempt to use the phone portion to make a phone call, but terminated due to airplane mode being on.
Since then, I have not used my headphones (no reason to) and have reinstalled Meridian. Tonight while browsing teh web with my headphones in (Youtube), my music once again began to play on it's own.
It seems to only do this when my headphones are in. I do not have doubletwist installed.