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.
Would high been the same as tall?
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.
How to remember high- nopseumnida as for example we take mountain is high-san i nopseumnida so you say no to climbing it