Are professors paid during sabbatical?

The short answer is: yes.

Sabbatical is not a vacation but is rather a leave of teaching to focus on research and publication. At many schools it is not automatic but must be applied for and is granted at the provost's discretion.

Some schools will only provide six months of funding, but will release a full year of teaching, making the faculty member responsible for getting grants for the remainder, or effectively working at half salary.

Yes. When you're not paid, it's normally called a "leave of absence."

In a perfect world, yes.

I work in a 400 year old pontifical university in the Philippines. I was tenured in 2012 which is supposed to entitle me certain leave privileges (LOA without pay, sabbatical leave, study leave etc.) I was not only denied any of these entitlement, but was also fired when I applied.