Independence of frequency in sound waves? Why does the frequency of sound wave depend only on the source? Why is the frequency and not any other "quality" independent of everything but the source?
And that said, why is velocity and wavelength of the wave dependent only on the medium? 
And finally - in case of the doppler effect in case of sound, does or doesn't the independence of these quantities hold true? 
 A: Maybe it is better to ask the other way around. Why is the frequency the only property that is independent? Although, mentioning the Doppler effect, you already found an example, showing that this is not exactly true. Hence the frequency depends on the source-velocity relative to me, and my velocity relative to the source. So "independent" is tricky and the question somewhat vaguely defined. That the speed depends on the medium should be clear, when looking at textbooks on acoustics. The frequency is constant, because source and observer have the same absolute time. If the source sends a pressure maximum every 2 ms, this is the same for the observer. This would change, however, if you consider relativistic effects.
In short, the source can only fix output energy (amplitude) and time between two wavefronts, i.e. frequency. 
A: Frequency is an intrinsic property of a vibrating source. Waves (of a certain kind) can be differentiated only with their frequency and not their wavelength. It cannot be changed unless there's a relative velocity between you and the source. That's the Doppler effect, which is the apparent change in the frequency when observed with a certain velocity relative to the source. It applies to both sound and light with the only difference that we'll take SR's corrections for relativistic speeds near $c$.
The only reason that the velocity of sound waves differ with medium is only because they're just mechanical pressure waves. Hence, they depend on the properties of medium like pressure, density, elasticity, etc. This can be noticed in the Laplace-Newton equation $$v=\sqrt{\frac{\gamma p}{\rho}}$$
(It should be noted that as you increase density, you don't just decrease $v$, but you're also increasing the pressure. Hence, the velocity changes accordingly)
In case of Doppler effects, you can also have a look at this question:
Doppler effect "apparent frequency"
