> Complicating the matter further, employees may choose to work without pay
This is wrong. If they're furloughed, they aren't working. Yes, by law (passed in 2019) they will get back pay, but they do not have the option to work without pay. If they're not furloughed, they may not be getting paid but it's not by choice that they continue working, it's because they weren't furloughed.
Depends on what "work" is. In theory furloughed workers aren't working, and can't go in to their workplace, but many different sorts of people are government employees. Yes, an IRS worker isn't processing returns, but it's not like government scientists stop thinking about their research even if they aren't allowed in their labs.