There's no special humanity wide significance to having a city on Mt. Everest. And there is also no new science that such a city would enable. Lastly, it wouldn't teach us anything about our place in the universe.
Talking about trial runs for Mars, which of course will be far less exciting. But can we imagine doing even that much? It seems preposterous to put a city on a tall mountain - but we're already talking about another planet.
It's hard to say when and how, of course. It might not happen in any serious fashion until asteroid mining and space tourism became real industries, which should result in improved tech and much lower costs for building Mars colonies.
