Learn Swedish: Grannen fäller grenen på sin tomt. - The neighbor cuts down the branch on his lot.

Sin is a reflexive possessive pronoun. It is used because the lot belongs to the subject of the same clause, which is grannen (the neighbor). If we used hans here, it would mean the neighbor is cutting down a branch on a different person's lot.

What determines whether we use sin, sitt, or sina in this sentence?

The pronoun must agree with the thing being owned, not the owner. Since the word for lot or property is an en-word (en tomt) and it is singular here, we use sin. If it were an ett-word like ett träd (a tree), we would use sitt träd, and for plurals we would use sina.

Is the verb fäller related to faller (to fall)?

Yes, they are a classic verb pair! Faller means falls (the branch falls by itself), while fäller means fells or causes to fall (the neighbor chops it down). Swedish has several verb pairs like this where changing the vowel turns an action happening to a subject into an action a subject does to an object.

Why is it just tomt instead of the definite form tomten?

In Swedish, possessive pronouns like sin, min, or hans already make the noun specific. Because the possessive does that job, the noun that follows must stay in the indefinite form. We say sin tomt, never sin tomten.