Other mistake: In the opening scene on the airplane, one of the men tells Lt. Cable, "The captain wants to know if you'd like to sit up in the cockpit with him?" Buzz Adams, the "captain" referenced in this line, is only a lieutenant. Admittedly, "captain" is a standard aviation term for the pilot of an aircraft, but it would be unlikely to be used in this situation, since the term has a specific military meaning.
Other mistake: Near the beginning of the film, the "Bloody Mary" number begins while Lieutenant Cable is still on the airplane, approaching the island. A few verses later (less than a minute of screen time), Cable is seen walking onto the base. There is no way the plane could have landed that quickly. (Note: In the original theatrical release, the musical number was in two parts, with a substantial scene in between, thus giving Cable much more time to arrive. The mistake only exists on the cut made for home video.).
Other mistake: During the "Show" scene, Ray Walston's character comes to the front of the stage and the Navy / Marine emblems are on either side of him. During the song, he goes stage left and the Navy emblem is there but when he goes stage right there is the Navy emblem - when he returns stage left, the Marine emblem has taken its rightful place - guess they had extras for each and got confused during the scene.
Answer: See http://www.imdb.com/title/tt0052225/fullcredits#cast.Bloody Mary's singing voice is Muriel Smith, Lt. Cable's is Bill Lee. Mitzi Gaynor (Nellie) does her own singing. For most of these "on-location" musicals, the singing is recorded in a studio, so even the actors who do their own singing are, in effect, dubbed by themselves.
Myridon