Does air in the troposphere regularly mix with air in the stratosphere?

Answer 1

Yes it does, though it does so slowly and through a variety of mechanisms. One is simply through strong convection from thunderstorms that break throughs the temperature inversion at the tropopause, but this is a relatively small amount of air.

More importantly, there are circulations that occur throughout the lower stratosphere and upper troposphere that slowly mix the air. The basic driver of this motion is the upward propogation of low-frequency waves, such as Kelvin Waves, Rossby Waves, or gravity waves (they have extremely long wavelengths and are not something you can observe directly...in fact they're quite difficult to conceptualize, at least I think so anyway). When the waves break (upward) through the tropopause (just like waves breaking on a beach), they add both air and energy, in essence, to the stratosphere.

An example of the consequences of this is the strengthening and weakening of the Polar Vortex in the Boreal winter. This vortex is a feature of the lower stratosphere, but it has direct effects on the weather in the mid- and high latitudes of the northern hemisphere. A stronger polar vortex tends to keep that cold Arctic air circulating at the poles, while a weaker one would let it slip further toward the equator.