When Tony leaves the destroyer to fly to Puerto Rico, he leaves in helo #406. When he lands in Puerto Rico, he is in helo #740, a different helicopter.
Outside shots of the ship are clearly the USS Arleigh Burke (DDG-51); however, interior shots and close-ups show the crest of the actual USS Paul F. Foster, as well as the red-and-white checkered crest of DESRON 9, which is the real-life destroyer squadron out of Everett, WA to which the Foster was actually assigned.
The video on Abby's screen while she is playing The Immortals repeats, showing she is not actually playing the game.
The destroyer in the show is referred to as the Foster, yet the hull number (51) of the ship is that of the USS Arleigh Burke.
The range of the helo shown (UH-3H) is 621 miles. Since the ship was shown just to the north of the Bahama Islands and was coming from Roosevelt Roads, Puerto Rico that is just over 1000 miles. Tony would have to fly to the States and then by plane to Puerto Rico. Also UH-3's are carried on aircraft carriers or shore based. They are cargo/personnel/VIP transport helos.
Crew size of a Spruance class destroyer is 19 officers and 315 enlisted men without an air det. When DD's carry a helo on board, there will be a support detachment (det) on board. So Gibbs answer on crew size is pretty close to the Spruance class not the Arleigh Burke class.
USS Foster is a West Coast DD. She never deployed to the East Coast.
DDG-51 is the Arleigh Burke and does not carry a helo, no hangar. A helo can land on the ship. She is home ported in Norfolk, Va.
After the opening credits, you can see the dead sailor moving a bit the eyebrows if you pay attention.
Abby refers to the main method of combat between the two players of "The Immortals" video game as "parley". She should have used the word "melee". Parley would have meant that they merely hurled insults at each other. Melee refers to some form of hand-to-hand combat.
Dr. "Ducky" Mallard admits to not understanding how someone would be interested in participating in an epic role-playing fantasy game like "The Immortals". Yet he is a renaissance man that quotes classic literature. He was undoubtedly familiar with "The Odyssey" and other classics where mere mortals challenge fantastic creatures. While he may have no interest in playing such a game, he should have understood the appeal of the game on some level.