The basic rule is that the artist owns the work. In cases of work for hire it depends on the specific agreement. It's possible to make an agreement where the financer owns the work, and one where the artist owns it and can resell it to others.
In the case of kickstarter the artists usually owns the work when it's done.
Fan art is normally thought of as plagiarism or derived works. It generally isn't allowed but the owners of the original work is of course free to allow it as they see fit, which is why fanart can sometimes exist. Exactly what goes depends on the whims of the owner. Thats why fanart is often allowed on a non-commercial basis only.
Some countries have copyright laws where the artist always have the artistic rights, and can't trade them away. Only the commercial rights can be given away. But that gets tricky in some aspects, especially when it is unclear who the actual artist is, for instance in cases where the work is the combined effort of many people.