Well, if you were at the speed of the Universe's expansion, they would become tail lights!
1) The photons, unable to escape, overheat the headlights and melt the cheap plastic mountings.
2) I get another @#$%^&! ticket.
3) I'd already have my lights on. If I get c out of my 6-cylinder Impala, I want people to notice.
Ask Dr. Science! He knows! He has a master's degree in science!
I've actually seen Dr. Science perform this very experiment on a PBS special about ten years ago. Extremely funny stuff.
Interestily what relitiviy predicts is that you would see you tail lights function as normal. However the light from all the stars in front of you would be squished and you would they would see a shift to blue.Ligth from stars in the rear would appear to be streched out and would appear to be more red.
Unlike most star trek movies you would still see the stars appear to be relitively stationary. If astronomers are correct in saying that the the nearest start is many light years away that it would still take many years for you to pass that star while the surrounding starts would have moved so little you would hard pressed to notice.
The light is blue-shifted all the way to hogh energy gamma rays (relative to you) and you fry.