Learn Hardcore Swahili: Jino langu lina maumivu.

Swahili nouns belong to different classes, each with its own possessive concord. Jino (tooth) is in noun class 5, whose possessive prefix is la-. You add your person suffix -angu to get langu, not yangu (which is class 9/10).

What is the li- in lina?

The li- is the subject-agreement prefix for noun class 5 in the present tense. It comes before the tense marker -na:
class 5 subject prefix: li- + present marker -na = lina (“it has” or “it is having”).

Why do we say lina maumivu instead of using inauma?

Both express pain, but they come from different roots:

lina maumivu uses the verb kuwa na (“to have”) plus the noun maumivu (“pains”), literally “it has pains.”
inauma uses the verb uma (“to hurt” as an action) with the class 5 prefix i-
- -na: inauma (“it hurts”).
  You can say either Jino langu lina maumivu or Jino langu inauma, with very similar meaning.

Why is maumivu plural if I’m only talking about one tooth?

In Swahili, maumivu is a class 6 noun meaning “pains” or “ache(s).” Pain is treated as a mass or collective concept, so it always appears in the plural form with the prefix ma-.

What is the plural of jino, and how would the sentence change for “my teeth hurt”?

Jino is class 5; its irregular plural is meno. For “my teeth hurt,” you use the class 6 subject prefix ya-:
Meno yangu yana maumivu.
(“Teeth my they-have pains.”)

Can I rephrase using nina instead of lina to say “I have a toothache”?

Yes. If you want to focus on yourself rather than the tooth as subject, say:
Nina maumivu ya jino langu.
Literally “I have pains of tooth my,” which English speakers often render as “I have a toothache.”