Light in an accelerated system

Special relativity states that no, the light would not appear to move faster just because its source is moving, or because it's "traveling along with" the car. Light always appears to move at velocity $c$.

If the car is traveling at close to $c$, this means the observer at rest doesn't agree with the car's driver on how long it takes light to travel the length of the car from back to front. This seeming paradox is eliminated by the fact that time is slowed down in the car. (The car's length is also contracted, which complicates things a bit).

We know that c is the absolute speed limit for anything. When we try to accelerate a photon as you have asked, another effect takes place. This is called blueshifting (or redshifting depending), and this is because instead of speeding up the light wave, it simply increases the frequency by effectively squashing the waves together. It is called this as it pushes the light towards the blue (higher frequency) end of the spectrum. Obviously, redshifting is the opposite which is when the photon source is moving away from the observer, and the light is shifted towards the red, lower frequency end of the spectrum. The wikipedia page on blueshifting is very helpful (https://en.wikipedia.org/wiki/Blueshift).