I've been working on trailer wiring for my boss. Hooked my truck up to the one stock trailer, and it was backfeeding power from the lights into the brake controller due to a broken ground wire. Fixed that and hooked it to the bale trailer, and now my left tail light won't light. The signal works fine. I think the trailer plug draws power off the left taillight. I'm suspecting the trouble could be in one of the many unprotected butt connectors I found under there.
So, it looks like this weekend I will be replacing and shrink wrapping butt connectors. Or I may just go to NAPA and get a fuse block and some relays. Run a 12 or 10 gauge wire back from the batteries and up under the bedrail. Then run power from my right taillight up and put it on a relay, for both tail lights, one for the trailer marker lights, and wire relays in parallel with the signals for the trailer plug.

Would a 12 gauge wire be heavy enough for the load? Or would I be better off with 10?

Would a relay pull enough current to run my flasher unit, if I wired the truck's signals off relays, too?