I do not know what happened but my previous reply was a failure. I'll try again
Ok, I checked the references you cited, particularly this: http://rockpile.phys.virginia.edu/mod04/mod34.pdf. Let us first make something clear. First, the Doppler effect mentioned in this reference is the classical Doppler effect derived from the relativistic limit when v less than c. Now, the calculations of the paper are made in the presence of gravitational fields. If we appeal to the equivalence principle, we are saying that the calculations assume a non-inertial system of reference (NIS). So, let's not mix things. In inertial systems of reference (ISR), in which space is assumed isotropic and homogeneous, the speed of light is always c relative to an observer at rest with the light source. If an observer moves relative to the source he will measure the relativistic Doppler effect. I agree with this. But for NIS the speed of light changes its values from place to place. To a certain degree, I agree with your result, namely
c'=c(1+gh/c^2)
but recall that it comes from an approximation. Actually, Einstein obtained this expression in his article of 1911 (take a look at my references). There he explains that the bending of light is due to the fact that the refraction index changes in a gravitational field (see reference 23 too), and therefore c is different in different points according to the above formula.
With respect to your question: c' = ? ; L' = ?, in his article Einstein arrived at the same conclusion as you, i.e. f varies and L remains constant. I can explain this as follows. Recall that the speed of a wave in a medium is not determined neither by the observer nor by the properties of the source but only by the properties of the medium. Let's consider that the aether exists. Assume then that a light source emits at a given f and with a given L. So, we would expect that the light speed remained the same everywhere at any time. Now, consider that the same light source is placed in an inhomogeneous aether. In this case f and L will remain the same relative to the source, but the speed of the wave fronts will vary from point to point as the wave fronts propagate. One can model this speed variation either as a change of f keeping L constant or, the opposite, keeping f constant and varying L (or both but in different proportions). However, this will create the prejudice that what varies is the frequency/wavelength instead of the properties of the medium. One has to be aware of this. Like I said in my essay one can give to this phenomenon different physical/mathematical interpretations, the general theory of relativity models the inhomogeneous space as a warped space keeping c constant.
I hope I have helped to answer your questions. Please work out the idea that space is a fluid.
Israel