I am going to comment on quite a few submissions made before.
1. There will be no Doppler shift between two objects moving at the same velocity whether you are using sound in a medium propagating sound or a medium (or non-medium) propagating light. There will be a Doppler shift on the transmit side, and the inverse at the receive side.
2. You would be able to set up an experiment such as the Michelson-Morley experiment for both environments. When you do this (remembering that we actually know that the speed of sound can be exceeded) you can now check to see if you can measure a speed relative to an "absolute rest" environment, which in the case of sound would be the medium.
However, what becomes suddenly clear if you use the sound environment, is that the sound going 90 degrees to your velocity, will not reach you when reflected. In order for the sound you send sideways to reach you, you have to send it slightly ahead of you, depending on your speed. If you travel close to the speed of sound in the medium, it will only reach you if you send it out close to straight ahead.
Apart from the distance increase of the path in the moving frame, you will therefore also have a Doppler shift in the 90 degree arm which will increase in relation to your velocity (wrt the medium), corresponding (either totally or to a large degree) with the shift in phase you are expecting because of the longer path length the light has to travel in an "absolute frame" when your velocity relative to this frame increases.
Interference will be in the "absolute rest" frame, before you measure your intervals. You may also have to consider the phase shifts and possibly Doppler effects at the "mirrors".
I have not checked these calculations, but have been lead to believe that you get a Null result for all velocities. Please do not quote me on this. I have the document where this is described by an 80 year old scientist, but have not worked through it. Do the calculations.
3. Assuming the constancy of the speed of light through the vacuum (or sound in a sound medium) is a fact (and we have very little if anything to doubt this), the question becomes if the speed of light in each reference frame is aconstant in both directions. The Lorentz transformation assumes this to be true and then just works with the speed of light c.
Now it is important to consider this: I can either assume the speed of light in both directions is c constant in any moving frame, and use the average of the time to an object and back to determine the distance, or I can assume that they are different, but that by assuming it constant I can have a model which describes mathematics for any frame without knowledge of my velocity relative to my "absolute rest" frame, bearing in mind that I might get some discrepancies.
The part I struggle with most when assuming that the actual relative speed of light is the same in both directions, is the Sagnac effect. The Sagnac effect is used in (amongst others) laser Gyro's and is also needed in keeping the GPS system up to date. In gyro's it is used to send light in two directions to come back to the starting point, where the difference in travel time between the two pulses are measured, when the gyro is rotating.
I have read that SRT explains this effect just as well as other models because the "extra" length causes the time difference. If I evaluate a small straight section of the gyro, I can however not find how to explain this if the assumption is that the constant c is the same in both directions. The velocity of the environment is constant. There should therefore not be any path length difference between the directions the light is traveling in according to SRT.
4. I want you to understand that I am not claiming that it is possible to move faster than light. If atoms are actually held together through EM fields (as we believe currently), and there is a "preferred absolute reference frame" (which most mainstream modern scientists do not believe), the atoms may actually fly apart when approaching or exceeding light speed.
I am saying that the Lorentz transformation prevents traveling faster than the speed of light, but that the mathematical models we use to represent the world has restrictions and assumptions (believes) built into them. We have to try and understand those restrictions.
We humans have a tendency to restrict ourselves with our own knowledge or beliefs. When we "know" it is impossible to go faster than light or for an object to be heated by a colder object or for energy to be created, we tend to overlook opportunities.
The more knowledge we accumulate, the more we tend to restrict ourselves by what we know. Most of the times we do not even realize that we have made a subtle change to what we assume (believe), and that that change is restricting us.
5. I am not trying to start a new belief or theory.