Carriers are used as an offensive weapon by the AI, unfortunately. They do build them, given the right circumstances. And I have had them attack my cities as if they were a destroyer or cruiser. They just died quicker, LOL. Carriers will also be used for naval superiority (!!) at times, and lose against all but ancient ships. I've never seen one attack an AEGIS or Battleship, however, and don't think I've seen one attack a Cruiser.
I can't recall seeing the AI use airpower on carriers, but I do suspect that if the AI had an aircraft "out" that they would use the carrier for landing if need be. But there is no strategic use of carriers I have ever observed.
Cruise missiles and nukes are handled a special way (other threads talk about that) that seems to preclude their being deployed on ships. I've never observed a AI nuke deployed on a sub, either.