2003 chevy 1500 rear brake lights not working, but over the cab break light is. bulbs look good, the running lights work.

Sounds like a bad turn signal switch . both side brake lights run through it but the third brake l;ight runs direct from the brake light switch .- play with the hazard button a few times and if the start working you need the turn signal switch

May be a bad bulb or its connection is bad, replace the bulb and grease the socket, under the back of the vehicle, check for dangling wires, if your emergency lights all work it could be in the turn signals, this has nothing to do with dome light, Matt