I think you have multiple issues here, most likely unrelated. Also, your truck does not have a GEM. The Trailer Disconnected message I assume occurs when the trailer is connected to your truck? You already performed some basic checks and also removed the add on brake controller. The best way to diagnose this is to test the factory TBC system with a brake emulator using a nice tool from Tekonsha that contains a trailer brake emulator and lighting circuit tester indicators. You can leave the tool in the plug and drive around to see if the message stops appearing. If it does it indicates a problem with your trailer brakes or trailer wiring. The tool if I remember correctly was around $40 but I bought mine some time ago - Google it to source it.
Tekonsha - Emulator
Tekonsha - Emulator