Do Wizards get to choose which animal their Patronus will be? If they don't get to choose, what determines their Patronus animal? EDIT - I didn't recall that there were any duplicate animals among the ones that were mentioned.
There is no indication in the books that a conscious decision is made. We see a significant number of students find their Patronus throughout the series, none of whom are noted as making a decision. There are also cases of duplicates - both James and Harry Potter are noted as having a stag as their Patronus, for instance, and Snape and Lily share a doe Patronus.
The closest indication we get of significance in the Patronus itself, other than some unconscious desire, is animagi - the two cases where we see both form and Patronus, they are identical: James Potter and Professor McGonnagal.
No, they do not. It is generally an animal that represents them, or a person they care greatly for.Harry's was a stag which represented his father Serverus' was a deer which represented Lilly
Professor Granger, in Harry Potter and the Cursed Child, confirms how a wizard's patronus is selected.
HERMIONE: How dare you! Fifty points from Gryffindor. And I assure you if anyone interrupts me again it’ll be a hundred points . . .
She stares around the room. No one moves a muscle.
HERMIONE [continued]: Good. A Patronus is a magical charm, a projection of all your most positive feelings, and takes the shape of the animal with whom you share the deepest affinity. It is a gift of light. If you can conjure a Patronus, you can protect yourself against the world. Which, in some of our cases, seems like a necessity sooner rather than later.
I would guess that theoretically, if you became deeply disenamoured of a certain type of animal (for example, if you loved cats, but then for some reason started to really dislike them) then it stands to reason that your Patronus would change.