I'm talking a shot in the dark. You have a loose connection in your steering wheel column. What you said about the steering wheel has everything to do with your problem.

 

Yep, the side brake lights are operated VIA the multifunction switch. The MFS performs stop lamp inhibit internally (rather than a separate relay) when the turn signals are on. Check the White/Tan wire coming from the brake lamp switch TO MFS for power with the pedal pressed. Check it AT the MFS. The reason I say that is because that same wire powers the center brake light, but there is a splice before the MFS. If there IS power there then you need to looking at your MF switch and the wiring going to it for signs of melting or burns.
YES if there are wires burned or damaged under the column that would mess with your headlamps. The MFS also does high beams.