If a mountain was over 10-12km tall (in the Stratosphere) would the peak of the mountain never be covered in snow (assuming that if like Everest, this mountain was once fully inside the Troposphere and the accumulated snow on its peak was melted by some completely unrelated event) The idea behind this being the stratosphere begins at 10km above sea level and is said to be free of all weather phenomena. Also the idea that there isn't a mountain that reaches the stratosphere, is that by design/effected by the laws of nature, is their some sort of limiting factor like the structural integrity of the stone upon reaching a certain height?

Would the parts of the mountain above 10km be forever free of snow while the parts below be snowed on?

Would having a massive chunk of rock poking into the stratosphere modify/effect the climate in some way?