Photon: speed and mass Is it correct to think that the speed of light does not depend on the speed of light source because photons have no mass, so they have no the kind of inertia that is associated with mass, so they can not "feel" (acquire) the speed of the light source?
 A: the photons will travel at the speed of light relative to both the moving light source and an object in another frame of reference. time dilation will bridges the gap so that both may co exist.  
A: 
Is it correct to think that the speed of light does not depend on the
  speed of light source because photons have no mass

In a certain sense, yes.
The Lorentz transformations guarantee that the speed $c$ is invariant; an object with speed $c$ in one inertial reference frame (IRF) has speed $c$ in all IRFs.
But an object with invariant mass $m$ cannot have speed $c$ in any IRF since, in that case, the particle's four-momentum
$$\mathbf P = \frac{m}{\sqrt{1 - \frac{v^2}{c^2}}}(c, \vec v)$$
would be undefined (when $v = c$, the denominator of the fraction is zero and division by zero is undefined).
However, (hand waving argument ahead...), if we examine the limit as $m \rightarrow 0$ and $v \rightarrow c$, it appears possible that there are zero invariant mass entities with speed $c$ and defined four-momentum.
And indeed, we find this to be the case for photons.
A: Is it correct to think that the speed of light does not depend on the speed of light source because photons have no mass, so they have no the kind of inertia that is associated with mass, so they can not "feel" (acquire) the speed of the light source?
No. Photons have an energy E=hf or E=hc/λ where f is frequency and λ is wavelength. The frequency and wavelength are there because photons have a wave nature. The speed of a wave doesn't depend on the speed of the emitter, it depends on the properties of the medium. Space is such a medium, with properties such as permittivity and permeability, wherein the speed of light is given as $c_0={1\over\sqrt{\mu_0\varepsilon_0}}$, see Wikipedia. This expression is somewhat similar to shear wave velocity $v_s = \sqrt{\frac {G} {\rho} }$, again see Wikipedia
As an aside, note that a photon has no rest mass because it isn't at rest. However it does have a non-zero "inertial mass". This is a measure of energy rather than a measure of mass per se, because when unqualified, mass is assumed to mean rest mass. See the last line of Einstein's E=mc² paper: "If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies". Radiation conveys inertia, which is why the photon has a non-zero inertial mass.  
