There must be a short in your wiring...Especially if you have trailer wires...I have the same problem on my Silverado....Haven't fixed it yet but narrowing down the possibilties...Good Luck
Brake lights should NOT be effected by the headlights. Brake light function is completely independent of the headlights. Assuming you are not refering to the tail lights, or marker lights, which are supposed to come when the head lights are turned on. Not having brake lights when the head lights are on is a big safety problem and should be corrected.
You do not need the key in the ignition to make lights work. They are "hot" all the time. The reason you Sid marker lights are coming on at the same time is because you have a bad ground in one of your brake lights. It's allowing electricity to jump from the brake lights to the tail light circuit. Take the brake lights apart, clean up all the rust and make sure they "ground" to bare metal. Problem cured.
By "exterior" lights I am forced to assume that your mean the tail lights, brake lights, license plate lights, and if available, the side marker and clearance lamps. When a trailer light wiring harness is attached to a towing vehicle's electrical system, it is generally tied into the tail/brake/license light wiring harness. If that is the case with your setup, then the same fuses that serve the vehicles light system[s] also serves the towed vehicles lights. Usually, one fuse serves the brake lights, and another serves the tail, license, side marker, and clearance lights.
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