You'd still be a vet, just one who works on beavers. Most uncommon animals just get care from a regular vet unless they're in really niche areas. For example my vet also cares for wallabies and kangaroos even though we're in the US. She just happens to be the vet in a small town where a family has them as pets.

Career wise, Vet school, then working at a beaver sanctuary or a zoo would be your career path if you really wanted to work with beavers in particular.

Unless you wanted to really specialize on beavers in an academic sense? In which case you'd likely be working in some kind of beaver research center, and you might be a zoologist or a wildlife biologist in addition to being a DVM.

Speak for yourselves. They get into ours even when it's up off the ground and green. We're <50 miles away.

This depends a lot on the local laws about artifacts. In most of the world it would belong to the landowner. Museums may try to strongarm their way into preserving it, but most of the time they have no legal authority.