Does natural satellite(s) of a planet affects its orbital velocity around a star? Please excuse me for a lay question. 

As we know, to calculate the orbital velocity, we take the mass of the orbiting body, the mass of the body being orbited and the distance between the two bodies into account. 

However, we don't really care about the natural satellite(s) (of a planet) in the calculations. Why are they insignificant? Why isn't the whole system involved in the calculations?
Yes, it does affect the planet's orbital velocity. Both planet and satellite move around the planet-satellite center of mass. That center of mass is orbiting the star with some constant velocity (let's assume circular orbits to keep things simple) but if you're only looking at the planet, then the orbital speed around the star will wobble up and down around the because of its additional motion around the planet-satellite center of mass.

For most planet-satellite systems this effect is very small since usually the satellite is very, very light compared to the planet. Only when you have a genuine double system like Earth-Moon or Pluto-Charon does this effect start to matter. For transiting exoplanet systems they're trying to infer the existence of exomoons by exploiting the effect this will have on the transit timing of the planet. So far no exomoons have turned up though.