As Bond said, he doesn't really have a true receiver, and it was only designed for a passive sub. So he's basically taking some speaker wire from the DVD player to a homemade RCA connection to get sound to the sub.
My guess is there's probably an issue with that connection. If it's a hum, I guess it could also be a ground loop (is everything plugged into the same outlet?). These DVD player "receivers" are usually set up for small, low impedance (2-4 ohm) speakers, as well, so using the speakers/sub you have (I would guess 8 ohm) may be causing some issue. Maybe the amp is burning out. You should really consider getting an actual receiver... you can get one for under $200 that will work far better than what you're using, and you already have all the speakers and a sub.
But the sub isn't shutting off because the Auto-off feature works by shutting down when there's been no signal to the sub for 10-15 minutes. If you're getting a hum, the sub is probably seeing that as a signal, so it will never shut off.
EDIT: After reading again, I think what you mean is you get the hum only when the system is shutting down, not all the time or the whole time it's off. That wouldn't be a ground loop, and probably isn't an amp issue, either. Most likely the solder connection.