Where can aircraft carriers carry out air operations? I'm interested in any military or civil airspace regulations about where aircraft carriers are allowed to launch and retrieve aircraft.

For example, are fixed wing aircraft allowed to take off from an aircraft carrier while it is docked in port?

I would imagine (though could be wrong) land-based ATC and airspace conflicts have to be managed, amongst other issues.
I don't know what permissions are required (guessing a fair bit of paperwork, involving not only the FAA but also port authorities), but taking off in port (in this case, a museum ship) has been done.  The USS Midway was stationary, looks like minimal wind, but on the other hand an acrobatic plane with lots of power. (Youtube)