Part 1: There is no back up light in a standard type tail light housing. If you want back up lights you will need a separate light. The 2 wires in your tail light housing are for brake/turn light and tail light. You need to determine which is which.

Part 2: Sounds like a ground problem. Grounding cab to frame may solve it as fenders may be isolated from frame ground but are bolted to the cab. If that doesn't solve it, ground the inner fenders to the frame.

"It's just a phase. He'll grow out of it." Mama, 1964

1956 Chevy 1/2-ton 3100
1953 Chevy 6100 "The Yard dog"
1954 GMC Suburban Now with a new proud owner.