[Carl]

This will sound stupid but I'm no Astro Phyisist (or Speller)
I woke up thinking this last week and no one has answered it yet and I'm looking for a simple explination.

I'm being chased by an enemy space ship. We are both travelling at the speed of light and for arguments sake we are 500m apart.The enemy turns their headlights on to see me better (it's dark out in space).

Question 1. Will the beam of light catch up to my ship, does the light travel at the speed of light plus the speed of the ship?

Question 2. When i look back at the enemy ship will I see the headlights?

Question 3. The enemy is carrying a standard issue rifle that shoots a bullet at approx 4500ft per sec. Will the bullet hit me. Is the bullet travelling at light speed plus the velocity of the bullit or only at the speed of the bullet.

Is the above the same principle as throwing a ball forward in a train or does it all change in space and at light speed?

Do the headlights and bullet theoretically travel faster than the speed of light?

I'm just a mug backyard astronomer but I cant get this off my mind an it's anoying the people i work with.
can anyone explain in plain English.

Regards
(not so bright)Carl