You need a special adapter harness to connect a trailer. Most trucks today have that built into the under-hood fuse box but most cars do not. The problem comes from the manufacturers changing the design about ten years ago to really complicate the issue unnecessarily for owners. Years ago you had one or two tail lights, and you had a brake light that was also the turn signal light. Trailers were built to match that. One bulb had two filaments inside, meaning two bulbs in one. One was the tail light and one was the brake / signal light.
Today almost all vehicles have separate brake and rear signal bulbs meaning there's three circuits on each rear corner, but there's still only the two circuits on the trailers. You can still wire the tail lights as before, but which circuit do you connect the trailer's brake / signal bulb to? If you connect it to the vehicle's brake light, it won't flash with the signal.
I'd have to see a wiring diagram showing how your trailer connector is wired, but it sounds like there's a blown fuse for the left tail light and that bulb is being back-fed through the mixed up trailer wiring.
If the vehicle's tail light is dim when it turns on with the brake pedal, that's a sign of a bad ground. Current goes through a bulb, can't find a path to ground, so it finds an alternate path through a second bulb. Both bulbs will light up but both will be real dim.
I'd start by disconnecting the trailer connector, then getting the tail light working again, THEN we can continue on. Look under the hood in the fuse box to see if there is any reference to a "trailer tow" relay, "trailer left turn" relay, or something like that. If there is not, the dealer will have the adapter harness needed that plugs into your vehicle's harness and has the wires that go to the trailer connector.