I don't know the answers to most of your questions. The brakes on my 1976 trailer are straight up electric magnet on drum brakes. Any working controller I've tried so far works them just fine. The way I'd test your system is with a 7-way tester. I could eliminate your tow vehicle or identify a problem with it. Very fast and easy. Cost about $10 USD.
If the tow vehicle is fine, then I'd use a jumper light, a small 12 volt
battery charger, and a small multi-meter. I'd remove the umbilical cord, and install a new trailer 7-way plug. That will make me re-wire everything. While I'm at it, I'd install a new breakaway switch and test to make sure it works brakes on both sides.