The Executive Officer (XO) of the "Patrick Henry" is wearing Surface Warfare Officer insignia. By law, the Commanding Officer and Executive Officer of a carrier are Naval aviators.
Lt. 'Tuna' Medwick was only a Lieutenant and Harm congratulated him on appointment as squadron XO. US Navy fighter squadrons XOs - Executive Officers - have the rank of Commander.
Captain Ingles told Rabb to get off his boat when all Navy vessels are referred as ships.
Skates would never be alone on the LSO platform - LSOs work as a team to guide aircraft to land as shown. She would never face charges even though the whole carrier was under-manned.
When the Marine gets the verdict from the Member and hands it to the Judge, he is wearing his cover. Covers are always removed when indoors, and put on upon going outdoors.