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When any of the truck's four doors are open, a few strange things happen. The headlights may turn off and on. The lcd screen on the stock sterio flashes off and on. The apparent sound of operating relays under the dashboard can be heard.
It happens whether the truck is running or not running ... key in or key not in.
I'd guess a pinched or corroded wire, whatever voltage turns on your dome light or open door chime when the door is open is getting into the rest of your accessories. Because it affects the radio and the headlights, I'm going to guess that the issue is in the fuse box in the cab, or a wire bundle leading to it. Without seeing a schematic, my assumption is that this is the only place those components would tie in together. Check the wiring to and from the fuse panel for pinches, frays or any damage. Then look at the terminals on the box itself. Maybe a small leak in the windshield is dripping onto it?