The brake lights definitely shouldn't be on when you connect to 120V. The external running lights, turn signals, backup lights, and break lights all pull their power from the tow vehicle. Sounds like you may have tied the brake lights into the camper 12V system.
I drew diagrams/schematics for all my wiring. It takes a little time but once you put it on paper it usually makes better sense. They have helped me troubleshoot issues several times. I listed color and size of the wire going to each circuit so I can tell, in most cases, where a particular wire goes.
Until you are sure what connects to what you may want to take a picture with the cell phone so you can keep it straight.
Also, if you post pictures with your question everyone can see what you are seeing.
Sound like you have a fun project started.