I think the atmosphere freezes as it approaches cold temperature and creates kind of an icy containment wall.
In principle this is correct—in a colder atmosphere, water vapour does condense and freeze,
forming ice particles. But your notion that these ice crystals then coalesce to form some
sort of semi-solid "dome" isn't supported by any current evidence.
The icy shell might be cloudy, transparent, etc... It's probably thicker at lower altitudes to contain the majority of air density in the troposphere. Here's an article on Sun Dogs caused by the Sun's light refracting off Atmospheric Ice Crystals...
Yes, that too is correct. So-called sun dogs—like rainbows—are caused by refraction of the sun's
light, but in no way illustrate any other atmospheric phenomenon such as a solid ice plane or dome.
Within the shell / Within the ice cube, the atmosphere is layered by different densities from heavy to light as you move up in altitude.
Again, that's correct. As one's altitude increases, the density of the atmosphere decreases. But there's
no current evidence that supports this phenomenon being due solely to some sort of physical containment
such as an ice "dome". As the Earth's gravitational force is weaker farther from Earth's centre, at higher
altitudes, air molecules can spread out more, and the atmosphere becomes "thinner".
But near the Poles of earth, the air becomes thinner more quickly. The very thin stratosphere for example starts at 4 miles in altitude versus 12 miles at the equator.
The
tropopause can—as you suggest—indeed vary from around 23,000 ft to 65,000 feet,
but in actuality that's a relatively tiny difference in terms of altitude; we have planes currently
flying at each of those altitudes without any intrinsic problems. Commercial aircraft typically fly
between 33,000 ft and 42,000 ft whilst the highest military air-breathing engine airplane was
the SR-71 "Blackbird" with a 90,000 ft ceiling (until their retirement in 1999).
I'm not sure why - I used to think it's because you're closer to space near the northern edge but seeing as it happens in polar south that sort of negates the argument.
Air temperature, location, and the rate of temperature change (per 1,000 ft) affect the lower altitude of the tropopause (on the global model).