When I use my right turn signal, all three of the brake lights - including the center high-mounted one - flash along with it. I assume (since I'm doing this alone) that when I step on the brake it's normal except for the right turn signal burning steady along with the brakes. The four-way flasher lights up everything except the back-up lights and the license plate.
Since the right, left, and center brake lights are all tied together, it is normal for all of them to come on at the same time. That's what they do normally.
What is happening is that your right turn signal is connected to the brake lights. When put voltage on the brake lights the turn signal comes on and when you put voltage on the turn signal the brakes come on.
When you turn on the hazard lights, you are connecting the left and right turn signals together, so both turn signals come on when you touch the brakes, and the brakes flash with the turn signals. Since you also said you were previously getting the "lights on alarm", it looks like you were getting brake voltage on the parking light circuit as well. You never answered my question about the turn signals being separate, but judging from the diagram and what you just described, I assume they are.
So final diagnosis is if the brake light if about half the brightness on the right side, then you probably have a loose ground. If the brightness is the same, then probably the brake and turn signals are shorted inside your electronic trailer light adapter (usually required if you have amber or separate turn signals), or in the associated wiring. Maybe the cable got squeezed in the door or something similar.
