Thanks for the replies, guys! Keep 'em coming!
Firstly: I guess I should have been more specific with the questions (although I tried to be pretty specific!), because it's not clear if the people who selected the "Earth is in a distant cloud of stars..." answers may have been Round Earthers, that seems to be the arrangement in the "conventional" Round Earth model. So it's perhaps too early to say what the prevailing belief is amongst members of the Flat Earth Society. Please do leave further details in the comments if you get the chance!
Secondly, the link that StinkyOne provided does indeed state that
"the general flat earth belief is that the earth’s sky is surrounded by a sky dome the holds in the air and protects us", but so far nobody who has replied to the poll believes this, so it's a bit much to say that that's the
general belief. It certainly sounds like whoever wrote that page believes it though!
However (sorry for missing it before) the wiki
https://wiki.tfes.org/Stars says
"The sun, moon, and stars are all rotating around a central point over the North Pole...The stars in the night sky rotate around common barycenters above the earth just as the sun and moon do. From a location on the earth's surface the stars in the sky might seem to scroll across the night sky with Polaris at the hub.", and the animation below seems to show a cloud of stars rotating "as a single unit" around a common central axis. I'm assuming that The Earth is supposed to be at the bottom of that diagram? The text of the article does also describe "a swirling multiple system", not sure that I quite understand what that, or the attached 400-page NASA paper, mean, but it certainly sounds like the author of that page is describing "a cloud of stars hovering above the earth all rotating around a central axis directly above the north pole".
Did I understand that correctly? Would be great if someone could explain this in slightly simpler terms - Pete, sounded like you had some ideas? I'm particularly interested in what kind of star movements (or apparent movements) you'd expect to see looking up into the sky on a nightly basis - since I'm not gonna be around for millions of years I don't have any way of testing some of the more "long term", but it's pretty easy to look out of the window at night and see what's up :-)