This article seems to discuss the topic at length in clear terms:

https://www.epa.gov/ghgemissions/understanding-global-warming-potentials

Different GHGs can have different effects on the Earth's warming. Two key ways in which these gases differ from each other are their ability to absorb energy (their "radiative efficiency"), and how long they stay in the atmosphere (also known as their "lifetime").

In a very simple and handwaving picture: molecules can vibrate and rotate upon absorption of electromagnetic radiation. But these are quantized energy levels. The absorption spectrum, hence the number of possible absorption lines depend on the molecule.

For example O2 has two atoms. They can vibrate against each other resulting in one set of spectral absorption lines.

CO2 has already three Atoms. Hence you get vibrational modes for first C versus the O atom, for second C versus O atom and for C versus C atom. This gives you several sets of spectral absorption lines.

As well you get different numbers of rotational absorption lines depending on the demension of the molecule. Helium is 0D, O2 and CO2 are 1D, H2O is already 2D because the angle between the bonds is not 180 degree. Methan is an example for a 3D molecule. The higher the dimension, the more modes of rotation are possible.

In general more complex molecules have more rotational and vibrational modes hence they absorb a larger number of spectral lines. Hence they absorb more infrared radiation.

For more details and a more physical mathematical description have a look at infrared spectroscopy.

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Does size of a molecule play into how high in the atmosphere it can get? Like is it that the bigger ones get warmer and therefore rise like a box of legos or does the mass keep them generally lower due to gravity? I feel like I’m overthinking this and that the effects of molecular size to dispersion are negligible

I don't know. The only example I know is that helium and H2 are that lightweight that they can escape into space. Hence the molecular weight seems to play a role, but that's all I know.

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The atmosphere is full of wind, which keeps things pretty well mixed, at least in the troposphere.

In a hypothetical perfectly still atmosphere, the heavier molecules would tend to settle to the bottom. Even then, the air wouldn't be perfectly stratified like a layer cake, because there is enough thermal energy to mix the layers. But you would observe that, as the elevation increased, the concentration of heavier molecules dropped off faster than the concentration of lighter molecules.

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