Regarding gas separation by gravity, please read this article:
https://www.researchgate.net/profile/Sathish-Dhandapani/post/Is_gas_gas_gravity_separation_possible_What_conditions_and_restrictions_are_needed/attachment/59d640b679197b807799cdd9/AS:431900461408257@1479984502567/download/FULLTEXT01.pdf
As you will see, separation of gases by gravity actually occurs, and can even be exploited industrially. It is more evident when the two gases have very different molecular mass. Hence in air nitrogen and oxygen do not separate significantly, as their molecular mass are very close (28 and 32 kg/kmol respectively). But when one of the two gases is much lighter (helium has a molecular mass of just 4 kg/kmol), then the separation occurs quite significantly, given enough time and lack of mechanical mixing.
And I can endorse it by first hand experience: when young, I served my military service, for one year, in 1984, in the fire brigade (instead of serving as a soldier). During training we were given an anti-gas mask, which in reality has just an active carbon filter, and we had to cross a large building saturated with a mixture of toxic gases.
As Oxygen is the heavier, it tends to stay close to the floor, whilst toxic gases such as CO, being lighter, stratify at higher elevation. Hence we had to crawl through the building for crossing it safely. Guys who did not bear staying on the floor for 20 minutes crawling slowly, and who did stand up and run away, were often seeing collapsing after breathing the toxic CO, and then they had to be treated with pure oxygen or even in the hyperbaric chamber. After the second failed attempt, the guy was expelled by the fire brigade and sent back to normal military service. I managed to pass through the toxic gas building at the first attempt, luckily, and with no harm...