No they live on the interior of it, a small-scale Dyson Sphere as ANGRYWOLF says. Cocoon is hollow; that's how come there's a huge hole blown in the side of it.

The gravity thing is easily resolved by centrigual force, or by the fact that fal'Cie are magic.

Also yes a sphere is a structurally sound shape, and any sufficiently large body will eventually end up roughly spherical. Though I doubt that would hold with something hollow unless it were manufactured that way. That said given that Cocoon is low enough in Pulse's atmosphere that there is condensation raining off it, and it seems to be in geosynchronous orbit, I'm fairly sure we have to write this off to fal'Cie magic. No way something that big and that close could maintain geosynchronous orbit unless Pulse is spinning at a fairly horrendous speed, and seeing as it doesn't have a day-night cycle in the order of minutes...