I'm fresh from ATPL met so can answer '1'
As you climb from sea level the air cools at roughly 2 degrees/1000 feet until you reach the tropopause, the layer of the atmosphere that seperates the troposphere from the stratosphere. Above this point the temperature remains roughly constant until you reach a vast altitude.
As the air at the poles is colder (and therefore denser) than at the equator, the polar tropopause is lower (and warmer) than at the equator - typically around 30,000 feet and -45 degrees, whereas the average height/temp over the whole planet would be around 36,000 feet and -56.5 degrees. At the equator it can be as high as 50,000 feet with temperatures as low as -75 degrees.
Makes sense if you think about it. The height of the tropopuase is governed by the density (and therefore temperature) of the air beneath it, which is far higher over the equatorial regions than over the polar regions.
Therefore it actually makes no difference what height you fly at once you go above 30,000 feet over the poles, the temperature will be a constant --45 degrees and density (which is proportional to temperature) will therefore also be constant.