When Harm and Karma take their Tomcats into flight, the numbers on the planes kept changing. In different scenes there were at least four different numbers shown on the sides or wings of the two planes (104, 53, 212, 114).
There are three (at least) different hull numbers shown for the "Coral Sea". One shot shows the superstructure number with a 65, the hat worn by the CAG has CVN-43 (Correct) on it, finally there is a scene with a life buoy that has Coral Sea CVN-75. The Coral Sea was CVB-43/CVA-43/CV-43 and was conventionally (oil) powered. CV-43 is non-nuclear. CVN-75 is the Harry S. Truman which is a nuclear powered carrier.
The incident with the pilot is supposed to occur aboard the USS Coral Sea some time after the first Gulf War. The Coral Sea was decommissioned in April 1990, and the fighting to free Kuwait began on 16 Jan 1991.
The CO of the Coral Sea is wearing a Surface Warfare pin, which is incorrect. US maritime law and Navy regulations require that both the commanding officer (CO) and the executive officer (XO) of an aircraft carrier carry the rank of Captain and they must be either Naval Aviators or Naval Flight Officers. This is because the primary function of a carrier is aviation warfare, as such it is important for the two most senior officers on the ship to be experienced in Navy aviation. The highest ranking Surface Warfare officer on a carrier is going to be the Commander that serves as the ship's second officer and the Officer of the Deck for the Combat Information Center.