Why would it drain the battery? And which battery do you mean? The car battery won't drain because it is a switched outlet and goes off with the ignition. The ipod shouldn't drain just because it is on a dead usb port, but leaving the ipod (or iphone) on for a long time certainly might cause the unit to discharge.
I use an iphone and I just turn it off when I leave it in the car, but my wife just leaves her phone on and it always charges back up when the ignition comes on.
Can it stay connected ? Don't see why not - any battery drain would be negligible compared to sustaining 45mph on the motor alone
Should it stay connected while you're away from the car ? Ahem - you'd be leaving a valuable item in plain sight in the car, never a good idea Saying that though, when I visited a service station earlier I didn't hide the iPod.
One observation though - my 4th gen iPod Nano properly remembered what it played when I disconnected it after turning the car off but didn't remember when I disconnected with the car still on. Could be another bug in the iPod software (4th Gen isn't as robust as the older ones) but could be something with order of disconnection.
There shouldn't be any problem power wise, but I stopped leaving my iPod Classic in the car because it completely failed when I left work to go home on the second day I did this. The high that day was only 76 degrees. It was working at lunch, so my best guess is that the hard drive overheated during the afternoon. It couldn't do anything but power up; I turned it off immediately! My thinking was that the hard drive "warped" and couldn't spin. I figured that it would be OK when it cooled down, so about 9pm I went out and it was working. The battery was completely drained, however, and it was behaving very oddly as a result. It was fully charged before it stopped working, that's why I thought there was some mechanical problem with the drive putting an additional load on the battery! Once I charged the battery, there seemed to be no ill effects!
The risks to iPods and iPhones with solid state memory is much less, but they also warn against being left in hot environments (like inside parked cars). I solved this problem by starting a mp3 memory stick project. It's more work than an iPod, but it's been in the car for weeks now and working flawlessly. It's also very small and I can close the console access door over it, so no one knows that it's there. No temptation to would be thieves!