If you travel on car with nearly the speed of light and turn on the car headlights: will it shine in gamma light instead of visible light? If you travel on car with nearly the speed of light and turn on the car headlights: will it shine in gamma light instead of visible light?
 A: Only to someone outside the car; if they are in your path they will firstly be irradiated to death, and anything left will be vaporized in the ensuing collision.
Of course, if they are behind the car, they will see all the lights red-shifted, and they will survive!
A: The key point to bear in mind if you are perplexed by questions like this is that all motion is relative, and all the effects of SR apply symmetrically between two inertial reference frames.
When you sit in your car and turn on the headlights, they produce visible light. To a particle passing the Earth at 0.99999999999c the light from your headlamps appears to be gamma rays. If you were able to drive your car at 0.999999999999c past the Earth then when you turned on your headlights they would produce visible light, but that visible light in your frame would appear to be gamma radiation to people on Earth. The key point is that your headlamps don't change, and nor does their output, but how that appears to others does change as a result of the Doppler effect.
A: It is a relativistic effect so it depends on the relative velocity with respect to the light source.
Imagine that your car is moving close to the speed of light relative to some road (let us forget about the physics of your car and the road for this question). If you are inside the car, for you the lights are in the visible spectrum. For somebody still with respect to the road, let us call this person P, there is what is called a Doppler shift. P will not measure the electromagnetic radiation with the same frequency as you.
The Doppler shift depends on the relative velocity (direction included). If the car is moving towards P at relativistic speeds, then P may detect gamma radiation (frequency goes up, blueshift). If the car is moving away from P, the shift is in the other direction and P may detect low frequency radio waves instead (frequency goes down, redshift).
A: "Gamma light" is confusing. Light is the visible part of the EM spectrum, kind of glossing over variations in who can see what. Gamma radiation means different things to different people -- according to this

Astrophysicists define gamma radiation as any radiation with an energy above 100 keV. Physicists define gamma radiation as high-energy photons released by nuclear decay.

If you use the latter definition, no. If it's not coming from nuclear decay, it's not gamma radiation. Page 8 of this document has an intensity spectrum graph -- it looks like very little energy is emitted by a tungsten filament in halogen in wavelengths shorter than 250 nanometers -- gamma rays are around ten picometers, so you'd have to blueshift them quite a bit. According to this online blueshift calculator, it'll happen at 299,792,457 m/s, the speed of light being 299,792,458 m/s.
A: It really is very simple. The wavelength of "light" from the headlamps as seen by a stationary observer will shorten due to the Lorentz contraction effect. The formula for this contraction is relative length = l * √(1 - v²/c²) where l equals the original length (or wavelength), v is the velocity of the source, and c is the velocity of light. If we express velocity as a proportion of c, then here are a few results.
For v = 0.1c, l = 0.995
For v = 0.5c, l = 0.866
For v = 0.9c, l = 0.436
For v = 0.99c, l = 0.141
For v = 0.999c, l = 0.0447
On the electromagnetic spectrum, taking the wavelength of visible light as 550nm (nanometres), at v = 0.999c, this would reduce to 25nm, which would put it in the long x-ray area. Only when v was up around 0.99999999999c would the "light" be short enough to be classed as gamma rays.
A: You’re missing the key point:
It will depend on what materials the headlights are made out of !  Its filament metals, LEDs, etc. ?
What are they made out of ?
What wavelengths do they transmit when their atoms/molecules are stimulated electronically ?
