If you drove at the speed of light, in the dark, and turned your headlights on, would you be able to see

You couldn't travel "at" the speed of light but your question is valid enough in the context if we talk about "almost" the speed of light.
The answer is Yes, because you could consider yourself at rest and that it's the earth that's moving at the speed of light under you in the opposite direction. Speed is relative. So your lights would shine as normal and you'd see a car in front of you going at the same speed as you just as you normally would. However the road passing under you at the speed of light would look strange, best illustrated in some YouTube videos on the subject.
Your question is physically the same as asking how an asteroid coming towards you at the speed of light would look in your headlights when you're out driving normally. It's the relative speed that's important.

Yes. According ot Einstein the speed of light is the only thing that isn't relative, time and space contract so that the speed of light is the same for everyone no matter how fast they're moving. Thus if you were in a car travelling at 99.999% of the speed of light and you turned your headlights on, you would think the light from the headlights were behaving normally. Distance and time would both be contracted by the Lorentz factor but the speed of light doesn't change.

You cannot move at the speed of light, that is impossible.

And you didn't finish your question.