This is from the Physics FAQ article that I wrote 15 years ago:
If shorter wavelengths are scattered most strongly, then there is a puzzle as to why the sky does not appear violet, the colour with the shortest visible wavelength. The spectrum of light emission from the sun is not constant at all wavelengths, and additionally is absorbed by the high atmosphere, so there is less violet in the light. Our eyes are also less sensitive to violet. That's part of the answer; yet a rainbow shows that there remains a significant amount of visible light coloured indigo and violet beyond the blue. The rest of the answer to this puzzle lies in the way our vision works. We have three types of colour receptors, or cones, in our retina. They are called red, blue and green because they respond most strongly to light at those wavelengths. As they are stimulated in different proportions, our visual system constructs the colours we see.
When we look up at the sky, the red cones respond to the small amount of scattered red light, but also less strongly to orange and yellow wavelengths. The green cones respond to yellow and the more strongly scattered green and green-blue wavelengths. The blue cones are stimulated by colours near blue wavelengths, which are very strongly scattered. If there were no indigo and violet in the spectrum, the sky would appear blue with a slight green tinge. However, the most strongly scattered indigo and violet wavelengths stimulate the red cones slightly as well as the blue, which is why these colours appear blue with an added red tinge. The net effect is that the red and green cones are stimulated about equally by the light from the sky, while the blue is stimulated more strongly. This combination accounts for the pale sky blue colour. It may not be a coincidence that our vision is adjusted to see the sky as a pure hue. We have evolved to fit in with our environment; and the ability to separate natural colours most clearly is probably a survival advantage.
1) Yes, in the real world, only a tiny portion of the light scatters by the Rayleigh scattering. This may be reinterpreted as the simple fact that generic places in the blue sky are far less bright than the Sun. It means that the generic places of the sky become blue but the Sun itself remains white. For the same reason, distant mountains keep their color. Also, the distant mountains don't increase the amount of blue light from other directions much simply because the intensity of light reflected from distant mountains into our eyes is vastly smaller than the intensity of light coming directly (or just with Rayleigh scattering) from the Sun to our eyes. And even if it were not smaller, e.g. when the Sun is right below the horizon and the mountains are needed, we won't be able to easily distinguish that the blue sky actually depends on the mountains.
Rayleigh scattering is caused by particles much smaller than the wavelength, i.e. individual atoms and molecules, so it doesn't really matter which of them they are. The rate of Rayleigh scattering is therefore more or less proportional to the air density which means that a vast majority of it occurs in the troposphere, especially the part closer to the surface.
2) The changing atmospheric density only impacts the angle of the propagation of the sunlight substantially if the atmospheric density changes at distance scales comparable to the wavelength. If the length scale at which the density changes is much longer than that, the impact on the direction of light is negligible and calculable by Snell's law.
If you ever watch Formula 1 races, you may see some fuzzy waving water-like illusion near the hot asphalt. This is indeed caused by density fluctuations caused by the variable heat near the asphalt (well, a campfire could have been enough instead of Formula 1). However, in this case the direction of light only changes slightly because the regions of hot and cold air are still much longer than the wavelength (half a micron or so).
If you think about ways how to get density fluctuations comparable to the wavelength which is really short, you will see that the source is in statistical physics and the naturally fluctuating air density due to statistical physics is actually nothing else than an equivalent macroscopic description of the Rayleigh scattering! When you calculate the Rayleigh scattering, you may either add the effect of individual air molecules; or you may directly calculate with a distribution of many air molecules and the source of the effect is that their density isn't really constant but fluctuates. So these two calculations are really equivalent. They are the microscopic and macroscopic description of the same thing, like statistical physics and thermodynamics.
If the blue light manages to come from a direction that differs from the direction of the source of light, the Sun, then – assuming that the atmosphere doesn't emit blue light by itself, and it doesn't (at least not a detectable amount of it) – it is scattering by definition. To get a substantial change of the direction, you need small particles, and that's by definition Rayleigh scattering. So there's no other source of the blue sky than the Rayleigh scattering – although the Rayleigh scattering may be described in several ways (microscopic, macroscopic etc.).
Well, there's also the Mie scattering – from particles much larger than the wavelength, especially spherical ones, like water droplets. However, for the Mie scattering to be substantial, you need a substantial change of the index of refraction $n$ inside the spheres, which is OK for water. Also, the Mie scattering is much less frequency-dependent (because $n$ only slightly depends on the frequency, nothing like the fourth power here) than the Rayleigh scattering so it doesn't influence the overall color much. Not only during the sunset, some grey-vs-white strips on the clouds near the horizon are caused by the Mie scattering. The Rayleigh scattering really has a monopoly on the substantial change of the color.
Best Answer
Rayleigh scattering results in scattering inversely proportional to the fourth power of the wavelength. So, no matter what the composition of the scatterer is, the blue will be scattered more than the red, as long as the main process is Rayleigh scattering. If you want to have another color for the sky, this will need to be the result of a different process or processes. Or the light from the star may have a different spectrum, with blue light being very weak or missing. An example of a non-blue atmospheric color is the orange haze in atmospheres rich in methane, like on Titan or on the early Earth. Rayleigh scattering has nothing to do with this color. It seems to be due to chemical reactions produced by UV in the atmospheric methane, resulting in hydrocarbons emitting some orangy light. The red color of sky at sunset on Earth (or after volcanic eruptions) is due to a different type of scattering, Mie scattering. This happens when the scattering particles are much larger than a molecule, comparable to or larger than the wavelength.