How much did Izamura know about the state of the planet? The spinning globe in his office implies he (at least, it's hard to judge how much the "average citizen" knows) is aware that Earth is spherical and rotates. Does he know about the fake sky and the truth of the world, so the "sinners get swallowed up by the heavens" business is just a trick to scare people into not asking any hard questions about astronomy? Like: "Why do none of the stars move?"
Considering the Truman Show nature of the Aigans' world, shouldn't one of them eventually have found the edge by traveling too far? Even if it's all fenced off and forbidden, it seems much more likely that one of these people would have hit the wall before they hit the ceiling.
If we take it that the entire world takes place inside a globe then they would have been able to circumnavigate the inside of the world. So long as a blue sky and stars remain above them they it would have looked exactly the same as living on the outside.
It's very unlikely they could go around the whole planet unless they've somehow hollowed out that much. This map seems to make the most sense given what we see: http://goo.gl/XZXqjd
If someone found the edge, they'd think "Wow, the world is under a big dome for some reason." They probably wouldn't realize that the dome is actually underground and that everything is inverted. Besides, maybe someone did find the edge and they were simply executed before they could tell anyone. Aiga is a fascist society, after all.
What was the structure at the apparent centre of the earth that they landed on. It was kind of a Dues Ex Machina that saved them out of nowhere and was never explained. Was it what caused the invertion in the first place?
It was (partially) explained earlier in the film, during Age/Eiji's class - that it was the result of an experiment dealing with gravity that caused the inversion effect. Which is why the ones responsible were called "sinners" and were said to have been punished by being swallowed up by the sky.
As for the celing structure Patema and Age/Eiji landed on, it was likely the underside of a long forgotten facility on the surface world. Similar to the service duct that was directly below the Agian research center. It'd gone unused for so long that the Agians had forgotten it was even there, which is how Porta and Age/Eiji were able to sneak in.
Bear in mind that the structure was "down" for Patema, whereas the true planetary surface was "up", meaning the structure was toward the core of the planet. The fact that it was able to operate autonomously for several generations and produce enough heat and light to simulate sunlight implies that it was tapping into the molten core / geothermal energy for power.
I had the impression that the entire world of Aiga, including the "ceiling", was a massive construction project that was built for the sake of inverted people, after the Great Change. They wanted the inverted people to have a home that felt like earth, so they carved out a massive chunk of the earth and filled it with inverted material (rocks and plants and stuff). The "ceiling", which is technically the floor, is basically just the point where they stopped digging. It's covered with some high-tech stuff that generates a fake sun. (I presume it also generates rain and wind effects.) It was all built for the benefit of the inverted people. As to how it was built...I have no idea. Some sort of long-lost technology, maybe.
Where the heck ARE they? Because the Alien Sky in the final scene makes it pretty clear that they're not on Earth.
That's an exaggeration, the sky doesn't look "alien" as in "not from this planet", but alien as in "not ours". Having an ring of space junk or moon rocks around the moon doesn't necessarily mean that it's not planet Earth they're on.
I figure that it is Earth. The moon is our moon, and it has a ring of debris because the Great Change sent a bunch of rocks and stuff flying into the sky. (Maybe the anti-gravity thing only works on Earth's gravity, so the stuff can still orbit the moon as normal.) And if the moon seems larger than it should be, that's just the art style.
On the other hand, maybe it's not earth at all, and the whole point is that the characters still have a lot to learn about their place in the universe.
The letter shown briefly at the end of the film dates back to the generation that caused the disaster. It states "Earth" several times and gives geographical coordinates within a city in real-world Japan. It's Earth.
If (for example) Patema and Age were to have children, which way would they fall?
Their children would fall sideways, like in thisxkcd comic.
More seriously, it would likely depend on whether the mother ate mostly normal food or inverted food during her pregnancy. In the specific case of Patema and Age, since Patema would likely spend most of her time in the underground city eating normal food, their child would be gravitationally normal.