When Gibbs is speaking with Seahawk's Captain, the Captain says, "I'll increase security on our weapons systems and the reactor..." The two ships used to stand in as Seahawk each have multiple reactors. USS Enterprise (CVN-65) has 8, USS John C. Stennis (CVN 74) has 2.
When Tony goes onto the flight deck we can see an F-18 parked near the end of the flight deck. When we see the ship from behind the plane has become an F-14. The F-14 was retired from US Navy service in 2006 and would not have been on board an operational aircraft carrier.
When McGee is taking photos of the crime scene, he accidentally shoots camera flash into Ducky's face. In the photo we can see Ducky carrying scissors in the pocket on his chest, but when the screen switches to 'real-time' view of Ducky, the scissors are missing.
The USS Seahawk has the hull number CVN-65 (carrier vessel nuclear), as seen on the caps that some of the crew wear. There is no actual aircraft carrier called the USS Seahawk; this is actually the hull number for the USS Enterprise. The exterior shots of the ship are, however, clearly not the Enterprise but a newer, Nimitz-class carrier. In the first establishing shot in this episode, the hull number painted on the deck of the carrier is 74, corresponding to the Nimitz-class USS John C. Stennis.
When Tony finds lieutenant Evans, he checks his dog tags and confirms that it's Evans. If the man impersonating Evans on the Seahawk wanted to maintain his cover he would have taken the dog tags as well as the ID card.