I'm sick again today, but will manage to do this one reply, because the first part of the answer to your (several) questions is easy-peasy. All four of your buttons are genuine Military-issue US Navy buttons... from various time-periods.
It helps to know that the one you are worried about being civilian or fake, at upper left in your photo, showing the eagle's head facing toward the VIEWER's left, is from 1941 up to the present. According to the button-book by Alphaeus H. Albert, on May 14, 1941 the US Navy issued an order changing the direction of the eagle's head from previous versions of US Navy buttons. Yours is shown in that book as button NA-121b. Also, the book says if your NA-121b button shows remnants of being coated with brown enamel, it is a Naval Aviator's button.
About the other three buttons in your photo:
At upper right... having a backmark saying "Oehms Acme Hall / Baltimore" means it dates from circa 1894 to 1900, according to the backmark-dating book by McGuinn & Bazelon.
At lower left... having a backmark saying "Scovills & Co. / Waterbury" means it dates from 1840 to 1850. Yours is shown as button NA-101a1 in the Albert button-book.
At lower right... being a 1-piece brass US Navy button with the same emblem as your 2-piece one from 1840-50 (except for the direction the eagle's head is facing), means your 1-piece one dates to a very short time-period, just the very-early 1830s. According to the Albert button-book, the US Navy emblem showing an eagle resting atop a vertical anchor was adopted in 1830... and 1-piece US Military buttons were replaced by the 2-piece type no later than 1835. Your 1-piece US Navy button is shown in the Albert book as button NA-107a. It most likely had an "R&W Robinson / Extra Rich" backmark, which means it is NA-107a1.
