Brakes locking says you are not getting +12v on the blue wire when you put truck in reverse. No turn signals/brake lights say the two turn signal wires are not getting +12v either.I would start in the middle, which is at the trailer wiring harness connector. Have someone put truck in reverse and with a volt meter, measure voltage from the pin going to the blue wire on the trailer and the ground (white wire) pin. If you have +12v there, the problem is in the trailer wiring. If you dont have +12v between blue and white, I would look for a fuse first. If you find one that is blown, you need to check the trailer wiring to see what shorted and blew the fuse.You say your running lights on trailer are working. This suggests your trailer ground is OK. Since both brake/turn and reverse are not working on the trailer, it is doubtful that all of those wires have failed, unless one shorted and blew a fuse in the truck.This should not be very hard to track down.