Wow..... According to Einsteins theory an object approached the speed of light, it would also need an infinite amount of energy to keep it there, so engine power is an issue. Its a by product of E=MC2
But, things are now dependant on the relative position of the observer. If, say for example he was at the speed of light minus 100Kmh, there would be a beam of light coming out of the headlamps, but the speed of the photons would be 100Kmh. As so far its unproven to be able to exceed the speed of light... latest research from the Hadron Collider not withstanding.
With the gunshots, if the shooter fired in his direction of travel, the bullet would leave the gun at its normal velocity plus the velocity of the car... or jet... in relation to an observer on the ground.
If the shooter fired rearwards in relation to his direction of travel, the bullet would still leave the gun at its normal velocity in relation to the gun operator. If the jet was flying at the speed of the bullet, then the bullet would still leave the gun at its normal velocity, and the pilot would see the bullet shoot away. But to an external observer on the ground the bullet would have no velocity at all, and would hang in the air, until it ran out of kinetic energy and gravity took over.
But to a chasing fighter aircraft, the bullet would still be lethal, as he is travelling at the same speed as the plane that fired it, therefore sharing the same relativistic space.
In the last case, it would be entirely possible to be hit by your own bullet, if you were travelling faster than it, and also in a downward path.
Confused? So am I, but we did this in 'Igh Skool Physics 30 years ago.