Learn Finnish: Miksi heidän anoppinsa asuu täällä? - Why does their mother-in-law live here?

In standard Finnish, when talking about a third person (like "his", "hers", or "theirs"), you must use both the genitive pronoun (hänen or heidän) and the possessive suffix (-nsa or -nsä). You can often drop the pronoun for "my" or "your", but you cannot drop it for "their".

How do I know whether to use -nsa or -nsä for "their"?

It depends on vowel harmony! Look at the vowels in the root word. Because anoppi (mother-in-law) contains the back vowels a and o, it takes the back-vowel ending -nsa.

Since heidän means "their", why is the verb asuu in the singular form rather than the plural asuvat?

The verb always matches the subject of the sentence. The subject here is anoppi (the mother-in-law), which is one person. Even though she belongs to "them" (heidän), she is the one doing the living, so the verb takes the singular form.

Could I use tässä instead of täällä for "here"?