You have 2 completely different problems.
The taillight are controlled and powered from the TV. I would first check to see if you have power to them at the plug at the rear of the TV. If good there check the ground at the taillight fixture. If it is good there test on the trailer side of the umbilical cord and on back along the wire to the taillight till you find the open.
If the frig is working on propane it has 12 volt
power and the control board is working on the propane side. Check for 120 at the receptacle supplying the frig. The next thing to check is the resistance of the element. Disconnect one end to test across it. if that is good there may be a 120 volt fuse on the control board. If these all check OK the board may be bad.