"The mountain is high."
Could you say something like 산은 키가 큽습니다, or does that construct only apply to people?
키 mean height as in a person's height, so no, u can't use 키가. Also, 높습니다 means tall/high, so it would be better than 큽니다 which translates more to 'big', in my opinion.
yes, depending on context :) for example u would say "a tree is tall" and "a mountain is high", but u can use 높습니다 both times.