The mystery was solved. Just as someone here thought, the ground wire was switched with one of the hot leads. Took just a few minutes to fix. Hooked back up and drove off into the sunset. All working properly and well, and NO, the moho lights need not be on in order for the brake and signal lights on the Jeep to work, as we were initially advised. No issues with the towing of the new Jeep. Tows as well as the old one did.
I also purchased a Protect-A-Tow rock guard that I'll install this weekend. That looks like a very nice product, too.
Thank you, everyone, for your responses to my inquiry.
