My theory is that, as you move towards the edge, the temperature continually drops. Eventually, the temperature reaches the point where different gasses turn into liquid and simply precipitate out of the air. This liquid simply pools and turns into streams that flow back towards the warmer regions, where it evaporates into a gas again and the cycle continues. In this model, by the time you would reach the edge, it is so cold that there is no atmoplane.
Do you have any evidence for this?
Are you saying that it does not get colder as you move rimward?
Are you completely ignore everything we just said? There would be no reason for the oxygen to actually move to the outer rim, Absolute zero isn't cold enough to liquefy oxygen! Liquid oxygen has a density of 1.141 g/cm3 (1.141 kg/L or 1141 kg/m3) and is cryogenic with a freezing point of 54.36 K (−218.79 °C; −361.82 °F) It will liquefy at -183 °C. Meaning that If someone were to travel there, It would be so cold, they would most likely die. Plus, The only theory someone could work out on how the oxygen would get there is that wind 'pushes it down, which in turn would interfere with your 'aether wind', so Your theory is completely bunk. Now if someone can give a theory with some premise, That would be nice!
News flash for you; absolute zero is colder than both 54 K and -183 °C. 
Yes, btw absolute zero is theoretically impossible, and even If it did get that cold, The liquid would freeze instantly, meaning that the oxygen wouldn't be able to flow back to the mainland, also, still no explanation as to why the oxygen get there.
At the physically impossible-to-reach temperature of zero kelvin, or minus 459.67 degrees Fahrenheit (minus 273.15 degrees Celsius), atoms would stop moving. As such, nothing can be colder than absolute zero on the Kelvin scale. So You are saying simply moving away from the sun makes it so cold. The sun is 5,778 K, So unless you are saying the sun isn't that hot, It wouldn't be absolute zero.
TL;DR: Still no explanation for the oxygen getting there
It couldn't get that cold
Another news flash for you: absolute zero may be theoretically impossible, but approaching absolute zero is not.
Do you have any more straw man arguments and incorrect assumptions to bring up?
So, you brung up that absolute zero is theoretically impossible, but what about freezing the liquid? or how it gets that cold just by moving away from the sun, even though the sun would still shine in that area, and how the oxygen gets there?!?!
Also, an answer to your other questions.
1. Um, It gets cold when you move away fro the sun, it is not like it gets hotter
2. Easy, the more oxygen is produced, the left over oxygen gets pushed back to the edge of the flat plane, than freezes and returns to the mainland. This easily explains the life cycle of air and completely proves our theory. You can think of it as the weather patterns:
The rain pours down, filling rivers (Frozen oxygen thawing)
That water evaporates (getting pushed into the edge)
The evaporated water turns into clouds (Frozen oxygen)
It rains, and the cycle continues. (Melting)
There, if you need anymore proof or have arguments to debunk this, I'll be happy to here it.