Furlings are a race that's still alive during the time of the TV series. They kind of look like ewoks.
No, that's just an idea from one of the characters. (Re-used in a "what-if thought bubble.")
Nobody really knows.
It was a riff on the idea that everyone assumed they were furry when nothing was known about them other than their existence.
However, to a question "When are we going to see the Furlings", one of the producers answers "Who says we haven't?"