Yes, brain surgeons are working class. They earn a lot because they actually produce a lot of value. If you suffer from a brain disease, getting surgery to fix it is probably one of the most valuable things you can get (depending on the severity, it will be the most valuable thing). Additionally, it requires a lot of skill.

Yes, what the hell do you think the word "working" is doing there?
If you're work for a living, you're working class. Even if you get paid a big salary, you're still a person who works a job for it.

They're upper class, which is part of the working class because they earn their money by their labor, aka working.

If they earn their money for simply owning things and collecting interest then they're not working class.

A brain surgeon makes money by working on brains and therefore is in the working class; the person that owns the hospital makes money by extracting rents from the brain surgeon and therefore is in the capitalist class.

Upper class and middle class don't necessarily correspond directly with those terms.