Why is DoubleTwist launching and playing when I remove the headphones?
Sounds just like a simple bug in DoubleTwist's app, and you should probably send them an email reporting it.  Programmers have access to system "events" (for lack of a better word) and we use these events in conjunction with the state of our app to trigger certain behaviors.  Like, if I created an app similar to DoubleTwist, I'd handle the system event of you unplugging your headphones to check if an mp3 is playing, and then I'd immediately pause it so that my app doesn't embarrass you by blasting your music out loud every time you unplug your headphones.  They could be performing this type of check (when receiving an alert from the system that the state of the user's headphone jack has changed) and either reading the state of the headphone jack incorrectly or evaluating the "are we playing audio" state of their own application (or both).