"The mountain is beautiful."
These words generally can both be used interchangeably. However, 아름답다 is used more often to describe scenery, and 예쁘다 is used more often to describe people or things. 예쁘다 is sometimes written as 이쁘다, which has the same meaning but is more of a slang word.
Almost. Because 산 ends in a consonant it would be 산은.
The whole 은/는 이/가 topic is pretty complex and nuanced. It takes a long time to master.
Link below from TOPIK should help though.