Can someone tell me if this is a problem or not? My subwoofer is making random noise, even when no inputs are plugged in. I've even tried to unplug it and wait for it to shut down completely, then plugging it in again. Initially it is in the off-state. After a few seconds, it switches to the on-state and is outputting random noise (not at a super high volume, but it is audible across the room even at 1/4 or 1/2 volume). This is with NO input plugged in.

Yes, that's a problem. Take the speaker to your dealer for service, something in the electronics has become "unhappy". Easy fix for your dealer to swap out the electronics modules.