but why does the Earth's atmosphere scatter light of short wavelengths, therefore making it appear blue, and lets through light of longer wavelengths? Does the same apply for other atmospheres? Would anything about it change if there was, for example, neon in the air instead of nitrogen? Or what if it was pure oxygen, ozone, CO2, or chlorine?... (I could think of just about any gas) I am curious whether some atmospheres in SE with rather trippy colors are actually realistic or not. (What about the infamous sulfur dioxide atmospheres?

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To understand this, you must be aware of this phenomena in its entirety: the oxygen-nitrogen content of the atmosphere, the atmospheric pressure and temperature and the spectral type of our sun, and of course any dust particles present all contribute. This system operates on a mathematical theory called
Rayleigh scattering, as you may know. The scattering of incoming light from our sun from small particles (like air or water molecules) increases as the inverse fourth power of wavelength. Hence blue light, which has a very small wavelength, is highly scattered, about twenty times that of red light. Via this process, the blue light is removed, and uniformly applied to the color of the sky. Some red, yellow and green exists (when the sun hits at an angle during sunset/rise, more blue light is scattered, and we get our red skies) as well, since the sun radiates ALL colors equally. A more intense source of radiation like this from, say, a nearby type F or A star, would only intensify the effects caused by other filters that contribute to overall sky color, which are:
Dust: The Raleigh scattering only occurs with particles of a small size, and so does not apply to dust particles. With dust, the
Mie Theory must be used instead, with broadly similar details. It dictates that the larger the dust particles, the more light is scattered, and thus the more grey the sky becomes at lower altitudes. Smaller dust particles scatter more blue light (as we see on Earth), whereas larger particles scatter more red light (like we see on Mars). Even more complications arise from this when we realize that the color of certain particles tend to absorb their inherent color, so red dust absorbs and scatters more red wave-lengths.
Temperature and Pressure are interrelated aspects here, at least inversely. Doubling the atmospheric pressure will double the scattering of light, whereas doubling the temperature doubles the brightness (reducing scattering), as an example. On Venus, we can see this happening vividly, where the temperature and pressure is so high that all sky color is scattered and intensified into a yellowish-white:

The last component is of course the gas content of our atmosphere: nitrogen (78%), and oxygen (21%). Argon gas and water (in the form of vapor, droplets and ice crystals) are also present. These operate on Rayleigh scattering, as described above, with modifications from the Mie theory in regards to the pressure and temperature of the gasses. Oxygen scatters blue light very effectively since it's molecules are smaller.

As you can see here, the shorter end of the visible spectrum (red) is closer in size to oxygen and nitrogen atoms, they interact more and are scattered to a higher degree. If only the shortest visible light was scattered, the sky would look more violet than blue. The rest of the colors would be green, yellow, orange, and red in decreasing intensity. I believe that answers your question about an eclipse viewed from within a pure O
2 atmosphere.
You mentioned sulfur, and this is a good example for the scattering in action. At 1 atm pressure, blue light is cut to below human eye visibility in less than half a meter, and the red is gone in fifty meters. Hence planets with a predominantly sulfuric atmosphere would have a dim murky yellow sky. With chlorine gas, obviously green light is allowed more then any other color.
Volcanoes, how can minute amount (compared to the bulk of the Earth's atmosphere) of volcanic material cause the Moon to almost disappear during an eclipse? What happens to all the sunlight?
This is caused by the Mie theory at work, by the dust and ash particles absorbing visible wave-lengths.
What about lunar eclipses elsewhere in the Solar system? I know the Hubble telescope observed Galilean moons while they were eclipsed by Jupiter. Jupiter is monstrous and has a much larger apparent diameter than the Sun when observed from any Galilean moon. Is the center of Jupiter's shadow much darker than in the case of Earth? What about the color of the eclipsed Galilean moons?
What about the color and brightness of any moon in the Solar system? Observable from the parent planet or other moons? With or without telescopes?
I would use Space Engine for that
An'shur! It is uniquely suited to such hypothetical.