Not exactly, clouds can be found all the way to the tropopause, just under the stratosphere and that is about 12 km over the poles and 18 km over the equator.Mount Everest is definitively within the troposphere. But then, the higher you get, the colder it is and the less moisture there is in the air. At 29,000 ft, there is very little moisture in the air and it doesn't "snow" as such.
What happens is that the air that meets the mountain has to rise and, by doing so, cools down and produce rain or snow. It is the so-called orographic effect of the mountain that produces the snow on the top of it. But, mind you, it doesn't snow that often on the top. But snow doesn't melt easier either.