We can say that in vacuum speed of light is constant. But if the star is moving with a certain velocity does it add that velocity to the velocity of photons emitted out of that star?
No. According to special relativity's postulates, the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.
You may want to read the relativistic Doppler effect. It refers to the change in frequency of light due to motion of observer and/or source.