That's a little weird. I would check out all of the wiring harnesses on the body and look for damaged wires where something may be contacting the copper. Also, take a good look at the grounds in each harness. Make sure they are in good shape. For your reverse lights, I would guess that your reverse indicator sensor on the transmission is sticking. Take it out and clean it up. There may be 25+ years of build up on it.
Good luck! It sounds like you may have some frustrating work ahead of you.