Do the gases in the Earth's atmosphere affect the color of a lunar eclipse? As the sun's rays pass through the Earth's atmosphere only the red light gets through. Is this the result of specific gases in the Earth's atmosphere filtering the red light?  Does all of the nitrogen and oxygen affect the sunlight shining through it?

Let's say the Earth (or just about any earth-like planet) has an atmosphere made mostly of green chlorine. When the sun shines through the chlorine atmosphere and hits the moon during a lunar eclipse, would the moon appear green? Would different gases result in the moon becoming different colors?
From the way you have formulated your question, it seems to me that you are aware of the light diffusive process named Rayleigh Scattering and why it explains the various tints and colours of the sky depending on the Sun position. Other readers can refer to https://en.m.wikipedia.org/wiki/Rayleigh_scattering.

We are used to our atmosphere and as such often discuss the RS merely on geometrical bases, as I shortly did just above.

However, the intensity of RS does not only depends on the inverse of the fourth power of wavelength, but also on the size of the scattering particles and specifically on the square of the molecular polarisability.

As such, changing the nature and composition  of the atmosphere does effect the intensity of RS. An example is SO2, which tends to intensify the yellow tint of sunsets in polluted area.

This should answer the first part of your questions. Let us move to the second part, specifically using the example atmosphere that came to your mind, chlorine as a green gas, i.e. molecular chlorine Cl2.

In this case, the gas is not colourless. Different respect to nitrogen and  oxygen, chlorine does absorb in the visible part of the electromagnetic spectrum.
It absorbs the colours opposite to green in the colours wheel, to keep things simple as possible.

In an inherently coloured atmosphere the relative importance of scattering will be less or at least variable depending on parameters such as atmosphere thickness, density etc. Let me ignore RS in such a toxic atmosphere :)

What is important now is that when you point your eyes up to the sky, the green colour you see is not that one whose propagation is hindered, but rather the one that can propagate (given that we are ignoring RS). In other words, the light will be always richer of green colour.

To make the final answer simple let us look at the moon when the only light that hit it is that coming throughout the planet atmosphere.

With Earth, the Moon is bloody as for RS has removed the blue violet part of the solar spectrum.

A hypothetical change on Earth atmosphere composition will also have a minor impact, as we have seen that the polarisability of the molecules counts.

About Chloro, the planet having chlorine atmosphere. During the eclipse,  its moon will have a greenish tint, or likely appear green. In this case the moon is lit by sunlight depleted of the wavelengths absorbed by chlorine.

To resume, the colour of the moon is always that less affected by the planet atmosphere but the mechanisms and the final result differ. 

When the atmosphere is intrinsically colourless, the colour of the moon is somehow "complementary or so" respect to that of the daily planet skydome. When the atmosphere absorbs, the moon and the sky dome will tend to the same colour.

Note: I did assume that sunlight can pass the intrinsically coloured atmosphere.  Again depending on various parameters, this is of course not given. Most likely we won't see the Moon. We shall really care of our atmosphere. 

Also the intrinsic properties of the moon do not change. But this is even more trivial.