The reason it follows the music is because those bass notes draw a lot of current, causing the alternator to draw more power from the engine. If you can see an effect from 63 watts of brake lights, 500 watts of bass will do it, too. Note that many A/C compressors draw up to 5 HP off the engine. Modern electronic engine controls should compensate for the load, but not all cars are the same. As for the bounce, what you are seeing is the control system feedback loop frequency response, which is either in need of repair (see below) or wasn’t designed right in the first place (which I highly doubt).
For the poor idle regulation at a certain temperature, there is obviously something wrong with the warm-up system. Given enough money, the car will run fine at any temperature, but now it becomes a question of how much you want to spend to make it right. A decent mechanic can tell you which part needs replacement, but I would hazard a guess it is the mass air flow sensor that is fouled a little.
Don
