Deliberate mistake: On the Hawkeye's radar screen, the Bandits virtually stand still after switching course to defend target (after destruction of the enemies' runway). While the Daggers complete their way to the Uranium plant, destroy it, evade the SAMs and return (save for Maverick, who got shot down), the bandits are still in the same place. This is a deliberate error: If the Bandits had progressed with normal speed, the Daggers couldn't have possibly avoided a confrontation on their way back. (01:34:42 - 01:40:30)
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)
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.