Learn Italian: Mio cugino fa il pompiere. - My cousin is a firefighter.

While da can be used with verbs like lavorare to express a role (lavora da pompiere means he works as a firefighter), the most common way to state a profession is using the verb fare followed by the definite article, as seen here with fa il pompiere.

Could I just say Mio cugino è un pompiere instead?

You can use the verb essere for professions, but in Italian you usually drop the article, making it Mio cugino è pompiere. However, using fare plus the definite article is the most natural and common way Italians express this in daily conversation.

I remember we usually need an article before possessives like il mio. Why is it missing here?

This is a classic exception. You drop the definite article when talking about singular, unmodified family members. Because cugino is a singular close relative, it is just Mio cugino. If it were plural, you would need the article: I miei cugini.

If my cousin were female, how would this sentence change?

You would change the possessive, the family member, the article, and the job title to feminine: Mia cugina fa la pompiera. Note that another very common, official term for a firefighter in Italy is vigile del fuoco.