Please help, strange problems.

[Jerry]

If you will check the BCM like Jpbroad1970 suggested, maybe a simple reset (battery disconnection) would help?

Maybe is just a poor grounding. If you have a multi-meter you can check for yourself. One contact on the bulb holder has almost no resistance to the chassis. The second should have the battery voltage (~ 12V) when the brake pedal is pressed. 

What about the third brake light (it's a LED type) does it works? Sometimes they are connected to the BCM but I'm not sure how it is on J. It would be worth checking this out as well.

The third brake light doesn't work in the usual way of pressing the brake pedal, but it does come on with the others when the car is in reverse.

Interesting, the stop lights only work when you put in reverse?
This makes me think that there may be a problem with a short circuit on the wires maybe.
I would remove one stop lamp bulb and use a multi-meter to measure the voltage (between ground and possitive contact):
1. When you press the brake pedal. -> you shoud read ~ 12V
2. When you put in reverse. -> no voltage 0V
3. When you have in reverse and press the brake pedal. -> you shoud read ~ 12V

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Did you already checked the wires in the rubber hose coming out of the tailgate?
Maybe there the wires are damaged.