Two aspects of your questions aren't answerable. Special Relativity (SR) doesn't include any transforms which tell us how things would look from the point of view of a frame traveling at c with respect to us.
I can slightly modify your last question to answer it. Let's say the car was traveling at 0.99c with respect to us. Then you wonder whether 'logic' would say that light from its headlights was traveling at 1.99c wrt us. SR says, however, that the light is going at c wrt us. The problem is that the 'logic' that led to the 1.99c result included a few assumptions that seem right but aren't.
These false assumptions
1. That the distances between any two events (e.g. light leaving the car and hitting a sign) are the same according to different observers.
2. That the time intervals between any two events are the same according to different observers.
SR contains a complete set of rules (the Lorentz transforms) for calculating the time and position coordinates in one frame, given the coordinates in another frame. The rules don't match our intuition. For example, events that are simultaneous in one frame generally are not in another. This is not
the result of a failure to correct for the time it takes to see distant things. People have been making that correction properly for over 300 years. It's a genuine difference in perspective between relatively moving frames, a little like the way things look different when you view them from different angles, Neither perspective is more correct than the other.
(published on 07/09/2009)