Continuity mistake: When Maverick and Rooster steal the F-14 Tomcat, Maverick adjusts the throttle to move the aircraft out of the hangar. When he does, the wing sweep lever (marked by black/yellow diagonal lines) is all the way forward: wings fully extended. When Rooster says both runways are destroyed, Maverick flips the cover up and pushes the lever forward to extend the wings, but it already was. Back means the wings are stowed, forward means the wings are extended.
Visible crew/equipment: In the final flight sequence in the F-14, the reflection of the camera equipment is in the visor of Rooster, but not Maverick. The reflections are not that of the pilot's seat or the RIO control screens in front of the RIO, ie, Rooster.
Continuity mistake: After Maverick and Rooster are shot down, they go to steal an F14 from the enemy. After the crash Rooster has a wound on the right side of his neck - in several scenes afterwards the wound moves back and forth, from right to left.
Continuity mistake: At the end Hondo is getting the harness out to catch Maverick when he lands, Hondo is wearing his flight deck uniform helmet, flak jacket etc but when he greets Maverick seconds later when he lands his helmet and flak jacket are gone, and he's wearing his camouflage uniform he had on underneath.
Continuity mistake: The scene in the hanger where Maverick is introduced. When Admiral says "be seated" there are only 2 rows of chairs. When Hangman looks over to Rooster he is in a new 3rd row.
Continuity mistake: When Maverick calls Comanche, the sun is behind the Daggers. When they descend below radar a few seconds later, the sun is ahead of them. (01:31:16)
Revealing mistake: When Maverick is chasing the 5th Gen fighter in the canyon, it cuts to a shot above his left shoulder. You can see a Gold, Red, and White Flag patch reflected in the canopy. Neither Maverick nor Rooster are wearing flag patches during the mission.
Other mistake: In movie, Harvard and Yale are the crew of one of the dual F18. Harvard is wearing the Navy Aviator Wings on his patch, Yale is wearing WSO wings. Later, in aerial scenes, Yale is in the front seat of the plane and Harvard in the rear - the wrong way round. (01:40:00)
Answer: You're correct - when you see Hangman's loadout before he lands back on the deck - he still has all weapons on the rails - it should be that he is short an AIM-9 or an AIM-120 AMRAAM depending on what was fired.
In all shots of Hangman's F/A-18E after the final battle (especially in the close-up right before he lands on the carrier), it is shown that there is an AIM-120 loaded onto the right air intake weapon station. The left air intake station is empty - indicating either no weapon was loaded or an AIM-7/AIM-120 was fired.
You can see that all his missiles are still intact and even if one is missing it would be very unlike to only load on the right and not the left as well. But where he says "This is your savior speaking" you can see all missiles are still on the plane and unfired.