If light from a moving away object, such as a star, would appear red-shifted (and blue in case of approaching object), why don't we conclude that the speed of light is affected from that. Doesn't it mean that since light from a moving away object appear red, then the speed of light with respect to us has reduced (C-V) ?
The frequency and the speed are related by ##v=f\lambda## where ##\lambda## is the wavelength. With light ##\lambda## changes as the reciprocal of ##f##, so ##v## remains constant.
So, this must be true even in case of sound. So, does the speed of sound also remain constant in such case?
When a car passes you by, and you hear the pitch of its engine's sound change, do you conclude the speed of sound in the air changed at that moment?
Yes, the speed of sound depends on the properties of the air (temperature, density, velocity, etc.), not the speed of the emitter. Two different emitters, one stationary and one moving but otherwise identical, will produce sound waves at different frequencies and wavelengths, but the same speed.