There are so many similarities (Doppler Effect, independence of wave velocity from source speed etc..). Try moving in your car with music and ask you friend outside record it while you moving towards him (playback speed faster + “blue shift”) and away from him (slow playback speed + “red shift”).
No. In "STR" the speed of light is the same value no matter how the measurer is moving relative to anything else. For sound waves, the speed of the wave is measured relative to the medium through which the wave travels.
Consider a sound source and receiver at rest relative to each other. The source and receiver have coordinated watches and have agreed on when the source will emit a sound pulse. That way the receiver can measure the time interval between emission and reception and calculate the speed of the sound. If the air is still (no wind), the speed will be one value. If there is wind, the measured speed will be different. This is not true of light.
Also, the mathematical form for the SR Doppler shift is different from the Doppler shift for a sound wave, due to the independence of speed of light.
No, they can't. In fact sound waves would lead you to Galilean relativity, and in all the cases where it differs from Einsteinian relativity they would lead you to the wrong result. This is a catastrophically bad model for Einsteinian relativity.
As a simple example of where things go most horribly wrong consider a supersonic aircraft, and that the velocity relative to the ground of such an aircraft is the sum of the velocity of it relative to the air and the velocity of the wind.