I leave an ipod plugged into my center console. It's been there for a couple years, no problems. It gets unplugged once in awhile to synch it with my computer to get new music on it, but otherwise I leave it there permenantly. I know it's not always on/charging becasue I've looked at it before I started the car and the screen is blank, then upon starting the car the screen lights up, as if I just plugged it in. So, the answer to your question is, as far as the center console, it does power down when the car shuts down. Not immediately, but eventually it does.
Active Autowerke filter, Dinan underdrive pulley, Dinan full exhaust and software, 3.45 Differential, Springs, Brakes, Dinan/HRE wheels, 275/30-19 w/10mm spacer, 295/30-19 w/3mm spacer, painted front reflectors, 30% tint, Escort 8500i & Lidar Jammer, AngeliBright halos, Euro rear foglights, 3M ClearBra.