Interesting, the regulator emits a pulsing voltage so ti would seem the regulator is working causing the erratic movement , it would also seem the gauge is working ok & feeding voltage to the tank connector or the gauge would not have moved at all . The way it should work is 5v pulse is fed to the gauge first , goes to the sender & grounds through the sender completing the circuit , so if you have power all the way to the sender then that should be where the problem is .
The brake circuit is another mystery altogether , power is fed to the light at all times but ground either at the park brake pedal when pushed or at the proportioning valve in the engine bay if there is an imbalance in the system , so why would the brake system be grounding though a powered circuit ? There is nothing on the circuit board that connects to the brake warning light , it just pops into a blank hole with a blue power wire in & a second Black ground wire out .
Are you sure nothing melted under dash ?