Nope, I’ve never had this happen to me on my M1 machine, Bluetooth or 3,5mm headphones. If you don’t have another media app focussed or active, pressing a media control button/key will open Music iirc.
No. It is generated by the headphones. Some Bluetooth devices, when activated, will try to restart playing and they send a “play” command to the host which responds with the most recent audio app. Many car audio systems do this, too.