These problems are most likely a function of either 1) a loose headphone jack or 2) a corroded headphone jack. The Droid takes inputs from headphones and it sounds like yours is moving around on insertion and making contact with whatever turns on the music. This has been resolved by either replacing your phone under warranty or by inserting a small piece of folded paper between the jack and the phone housing.
The corrosion issue has been fixed by 1) turning off the phone, 2) removing the battery, 3) cleaning the jack with a pipe cleaner.
I have the same issue when I insert my headphones. I doubt that this is caused by corrosion as my phone is new... Issue started with stock 2.1 and occurs with Bugless Beast v0.4 Froyo 2.2 rom's. I have even tried different wired headphones. Does anyone know of a fix for this issue??? I have tried to remove the music app but the option is grayed out.