'02 Century BCM issue?

I have an '02 Century that my son drives. He said that two weird things are happening. 1) the clock on the digital radio stays on all the time and it is draining the battery overnight. Pull the fuse and the problem goes away. 2) the driver's side window goes all the way down every now and then when you start the car. It won't go back up for a while, but if you keep trying, eventually it will.

I've seen posts on either of these problems, but not both together. I'm wondering if this is a bad BCM?

Thanks