Dave:
The fleetside box on the '60-'62 is the same as the fleetside box on the '62-'66, with the exception of the rear cross sill. The mounting perches on the earlier fleetside boxes are attached slightly wider than the frame rails on the '63-'66. If the rear cross sill is good shape on the donor fleetside box, you can cut the spot welds holding the mounting perches onto the rear cross sill and mount in the correct location of the frame rails on your frame. I have a '60-'62 fleetside box on one of my '66 rolling chassis and this modification is necessary to make it work properly.
With regards to the factory back-up lights, they were a factory installed accessory starting in 1966. Prior to 1966, they were only available as a dealer option, per the factory manufacturer's literature I have.
A good used box is hard to come by and with the exception of the cross sill issues, will make a great candidate for your project. Good luck with your rig.
Tony