Thoughts on rear lighting
I would start by checking for the proper signals coming out of the PSDM.
Under the rear seat, locate the 2x2 molex plug that peels out of the main rear harness and changes to the rear Tail Light Harness.
A digital meter has a tough time showing pulsed signals (like a flasher), so use a test light to verify the flashing turn signals at the plug.
Your target wires are Bk/DkBl and Gy/DkBl (ref ChassGnd).
Vt/Db is going to be your RunLight wire.
Brake lights are using the same filament as the Turn.
If you have signals working at the plug, then you can move towards the back.
I have seen some strange back feed happen if there is no proper ground. This produces some really confusing symptoms.
Your pic showing 4 tail lights at the back of your car means that you may have a version of the New York Lighting/wiring. The upper Head rest lights were wired as Running Lights only (as in the following diagram)
(good catch JarJar). If that were my car, my OCD would make me run the extra wires up to the upper lights and make them work with the bed lights.
Weak Brakes
When you press brake pedal, how far down does it go?
Is the pedal firm? or kinda spongy?
Handbrake
Emergency parking brake also doesn’t activate the display light. Cart can be driven while parking brake is engaged without any warnings.
Key to this is that your car is a 2005. It is the first year of the Gen2 cars and one of the odd quirks of this year. The wiring guy didn’t quite have his act together and did not have the wiring installed to interrupt car function when hand brake is applied. It makes it real easy to drive around all day and smoke the rear brakes.
This may also be the reason why the park brake does not work so well. As JarJar suggested, if you have more adjustment in the cables, give that a try first. Otherwise, pull the drums off for an inspection to see what is left.
True- there is a switch at the brake handle, but it’s function only sounds off the beep when KeyOFF and BrakeNotSet (keeping you from walking away and your car from ending up in the lake).
Another part of the root issue is that the T3 controller did not have software programming that was concerned about the HandBrake pin13.
I suppose a creative wiring hack could add a few wires and a relay to fix this major oversight.