Trailer Lights Only Work When Truck Lights Are Off. When we but boat back on, nothing again. IF the trailer lights worked, but turned on the wrong lights when activated, would suggest that some of the wires were "crossed," miss routed.
The inside dome lights only work when the truck is running. Sometimes it takes a few blocks before you work off the rust in the coupler and on the ball. Your truck plug may or may not have an extra plug for a hot wire.
I turned the truck off and went into the house for several minutes to ponder this.
Your brake lights should work when you hit the brake pedal, and turn signals.
Once you know your vehicle connector is wired properly and works as advertised, the only source of problem. Ensure that the wire to the trailer is plugged into the connector on the back of your vehicle. Either one light shines more than the other or the taillights completely switch off when you press the brakes.