Black head and white head go to the factory radio. They go from the dash to the fader and then get distributed fore/aft. They say L and R on the back.

The square connector IS for the cigarette lighter AND the factory radio harness. They use the same connector. Look on the back of your ashtray.

Do your hazards work? I believe the signals get routed through there.

FYI: lights behind the HVAC switches are either bulbs on the HVAC CCU, or fiber optic off the octopus. If they don't work, it might be your rheostat or your cluster lights are simply turned all the way down.