2005 Brake Light, Back Turn Signal Struggle
 

I have a 2005 f150 XLT. The back turn signals are not working. Quickened blinker signal. The Hazards in the back do not work also. The brake lights are not working; the 3rd brake light in the center is working.
I've checked all of the fuses and replaced them just in case. I just replaced the turn signal switch. The bulbs are working. When I reverse the reverse lights come on, and the red lights turn on when I turn my headlights on.
I did have a new radio and a backup camera installed. The camera stopped working first. Then my left blinker went out about 6 months later, then my right blinker about 2-4 weeks later. I connected my backup camera wires, and now my radio will say "back up camera" but not display any video.
If anyone has any help on what I should check or do, I would be very appreciative.

This sounds like a grounding issue. I would check all of your grounds, especially underneath.

Looking at my 2004 F150 Wiring Diagrams manual, it could be G303 (ground), but not if the driver's rear-view mirror and door lock switch work. They use this ground too.

Could be both fuses 42 and 36 but you said you checked all the fuses.

I don't see any other shared power or ground for just the rear lights. So, I suspect the problem is related to the radio and/or backup camera installation. I would inspect their wires to see whether any were spliced into existing wires, and if so, I'd probably disconnect them and retest.

Hope you can find the problem.