Why build another Cocoon?

Because the current Cocoon is going to fall.

But why not use the gravity thingies to stop the old Cocoon from falling?

Because it's probably too risky to add them to a planet populated with many people - what if something went wrong and Cocoon fell? It's easier to put them on an empty Cocoon and get the devices up and running before people jump ship.

Why not just move to Pulse?

Pick your reason. People on Cocoon do not want to live there, Pulse Fal'cie, dangerous wildlife, and harsh conditions. Hope's reasoning: If we don't HAVE to force evacuation to Pulse, then why should we?

How is Cocoon sustained since the end of the Fal'Cie?

Human academics and their technology has replaced Fal'cie technology and accounts for the ecosystem on Cocoon.
It seems that not all areas of Cocoon are functional though due to the fact that some have moved to Pulse... particularly, the city of Bodhum was destroyed and recreated as New Bodhum on Pulse and then there is the inhabitants of Academia. What did people do before Academia replaced the Fal'cie? That is a plot hole, but I guess we can assume that they figured out how to survive during that time, perhaps with residual resources.

Well, that's my crack at it.