Electrical problems
 

Starting yesterday, as soon as I turn my truck off, and all the lights off, a strange clicking noise starts coming from under the steering column to the right. Then the compass/thermometer and rearview mirror adjuster turn back on and won't turn off. They stayed on all night last night. I realized they can be turned off manually but it's still a nuisance. Also the truck will also occasionally make very loud annoying beeping sounds while driving. And suggestions?

Your description is that of a GEM box going bad. You said it was the right side of the dash. Do you mean the passenger side of the radio or just to the right of the steering wheel? The GEM is by the left side of the steering wheel. There is a group of relays so it might be coming from there. I'm betting on the GEM personally. Good luck.