What's the actual structure/shape of Cocoon?

What's the structure of Cocoon?

Cocoon is a hollow sphere. It has a big hole from an attack from Pulse in the past.

Are people living on the outside of the sphere? Do they live inside (hollow world)?

The inside. If you look up in the "sky", you'll see what appears to be dense clusters of stars, but are actually city lights.

If so, where's the center of mass/gravity?

A wizard did it. Seriously, all you need to know about the gravity of Cocoon is on the inside, "down" = outward.

However, a bit later it's revealed that down there(?) is a giant ocean which is transformed into crystal.

Yes, Sanctum was transporting the Pulse Fal'cie over a large lake. That lake is on the inside of Cocoon.

To expand on the currently accepted answer, it's indeed a hollow sphere with people living on the inside (gravity pointing outside).

A later video sequence (Chapter 11) shows a very nice render of Cocoon, where you're finally able to see the crack properly (without misleading reflections of the sunlight from outside):

There's actually a rather huge chunk missing, it's not really just a small "crack". It's also possible to estimate the thickness of the outer shell, so it's actually not that thin.

I don't think it's ever explained what that transparent "cover" is, but I guess it might be the magical/artifical gravity field and/or water (which is also pouring out).

Besides that, it's really interesting how many different depictions of Cocoon there are in the game. The "game world" view is obviously lower quality and only shows a dark patch rather than the crack, but even in the rendered video sequences the appearance seems to change (even if only slightly).

The following image is from another video, which looks like there's water on the outside as well, but I guess this is actually inside and we just perceive the hole (part you're looking through) as "oceans":