It sounds like wiring problem, especially since other posts (from search results) suggest that wires going to the trunk latch can short and cause this. It seems like you have narrowed it down to your headlight (which means tail lights are on too) as well as the brake lights... so I would be searching for a short between this combo and the latch wires. The fact that wires to license plates were pretty corroded also leads one to believe there is wire shorting problems in that general vicinity.

I know, I just stated the obvious.. but sometimes it helps.