The U.S. Navy requires that, unless you are on duty, you uncover upon entering a building and once uncovered you do not salute. Several times in this movie a salute was rendered by an uncovered sailor.
Where the crew is listening to the aircraft radios attacking Korean targets they can hear the sounds of battle and the crash of a plane even though no pilot is talking. This would not be possible unless someone keyed the mic and left it open.
U.S. Navy ships at General Quarters (battle stations) do not permit smoking. In the scene in Combat Information Center Admiral Hoskins is shown lighting up.