Mine stay on for around 15 minutes, and as mentioned once the BCM goes to sleep. But you can't keep your key fob in the car or close to it, as I don't think the BCM will shut down.
Industry standard is based on being in the drivers seat for left and right side. This includes when buying parts for your vehicle, like a right outer tie rod end, servicing - it pulls to the right, etc.
My 21 Badlands has the wiring for the lights too, just need to install my new switch and install some lights. Also I will be working on the rear fog option.
Do you have the back up sensors in the rear bumper? That plug looks like the same as mine where those sensors are plugged in. The wiring color for the rear fog should be White with Blue Stripe for hot and Black with White Stripe for ground.