Do cats get haircuts?

Not generally, though some are groomed, and shaped for shows and such.
Some cats have health issues and have to be, not shaved, but closely cropped.

Depends on the breed and how deep down the rabbit hole the owner has gone...

Mine doesn’t, just her nails 🤷‍♂️

Depends on if the cat has long hair

Shave them bitches bald.

Sometimes.