"The house is small."
Yes. "집이 작습니다" does work. It literally means "The house is small". With regards to this example, the meaning is obscure, however. So, it could mean "The house is small" on the outside (overall size) or on the inside.
It should. Here they seem to mean small as in cramped, not much space. However, that's not clear from just 4 words.
What is the difference between 집이 작습니다 and 집이 좁습니다? Like, what are the situations they would be used in?
좁습니다 is in regards to how narrow or cramped the house is as opposed to 작습니다 which means how small it is in size
If the house is small on the outside, it is smaller on the inside. (Welch's first law of sizemology. To wit: Something can not be bigger on the inside than it is on the outside.