The flat earth is constantly accelerating "upwards", which is what creates the equivalent of sane people gravity. So it's not really falling over the edge as the Earth leaving you behind.
Interestingly, General Relativity shows that gravity is equivalent to the surface of the earth constantly accelerating upward, even on our lovely globe.
Yes, that's their belief. Does that mean that there is no gravity beyond earth? Also how does the sun work then? Do they believe we are geocentric, not heliocentric?
The sun and moon are tiny and relatively close. They rotate in circles above the surface at a constant distance (don't ask why). From what I know they spend a lot more time "disproving" existing theories than finding new ones to explain their model.
It's definitely one of the theories I've seen. Checking on The Flat Earth Society website I see it's not their official position (but they acknowledge some groups do believe that). They don't have a different theory though, just that "objects simply fall".