Electrical Issues

I've encountered a slew of strange issues with my 2012 GTI lately. Here's a quick summary:

1. Keyless entry started acting up (wouldn't work, wouldn't work on one side, gas filler cap would start unlocking itself over and over again when I got close to the car, etc.)
2. Battery started dying. I'd throw it on the charger and it would be good for a few days or if I went for a longer drive it would charge up but after a few days it would barely be starting
3. Weird noise when cabin fan was on. I sometimes her a weird whirring/spooling sound coming from the console when the fan is on level 1 or 2. The sound pitch changes with revs and goes away with the fan off.
4. Quiet humming coming from center console after engine is shut off. Very quiet (almost sounds like a computer fan), can only be heard inside the car, can be heard 30 minutes after the car has been turned off.

I'm not sure how much of this is related.. #4 could certainly be causing #2. I just put in a new battery and while I haven't had issues starting yet, all the other symptoms are still there and I'm worried I'm killing this battery too. Has anyone else experienced these issues or have any suggestions as to where to look?

Maybe your battery needs to be replaced?? Also, keyfob might need internal battery replaced.