Do Authors Work From Home When Writing and Publishing Their Books?

Most do, but many writers prefer to work away from home, perhaps in a shed in their garden, or in the local library, or (especially when they're getting near the end of a book) they book into a quiet hotel for a week or so.

Everyone is different. There are no rules; there's only what suits a particular writer best in his or her individual situation.A lot of people work well in cafes; some like to be on a train.

They could but they don't have to. 

Some authors work from home, but not all of them.

J. K. Rowling wrote the first Harry Potter book in a tea shop.

Most do. ...................

Nearly all of them do. There are some who, for various reasons, isolate themselves in a remote place where they are less likely to self-distract, but that's an indulgence most authors cannot afford, plus they miss their loved ones too much to separate for that long.

The fact that some authors do this is a plot device Stephen King likes, although I don't recall that he himself does it.

It depends on the author, but most of them do, judging from what I've seen in interviews.

Yes, it seems many do; more so now.