Are professors paid during sabbatical? I'm mainly concerned about schools in the US. Answers about other countries are also welcome.
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.
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.
In most universities in Australia, you are paid a full salary when you go on sabbatical. You will also typically receive an allowance that hopefully covers most costs associated with travel and accommodation. You may also get an additional allowance if you are travelling with family. That said, such allowances often do not cover the full cost of travel. In such cases it helps if you have additional sources of academic funds (e.g., grants, consulting funds, etc.).
From what I hear, academics in Australia tend to be eligible to apply for sabbatical more frequently than in other countries (often every three years). That said, obtaining sabbatical is not automatic. You would typically need head of department support presumably based on a strong rationale for the visit in combination with strong research or teaching performance in the preceding years.
Here's a few examples of policies: