1992 Toyota Corolla

Most likely a bulb is burnt out. Remove the plastic housing in the rear in trunk.

If you see any black/burn marks wires/plastic housing burns, you may have to solder the connections.

Over time the plastic will get brittle and break, the wires screw into the plastic housing, if the screw threads are stripped, replace the screws with machine screws and nuts.

There should be no relationship. Brake lights are activated by the brake switch on the brake pedal.

Inspext the wiring under the dash, there may be a loose wire shorting, also make sure the brake lights are properly grounded.

Have you had any recent damage to front? Any aftermarket add-ons? Stereo etc?