Sorry, but the brass 2-piece button showing an anchor with a rope (and nothing else) is not a US Navy button... because it has one of the classic civilian Fashion button self-shank backs. The US Navy did have a similar-looking button in the early-1800s, but it was a ONE-PIECE button, and its anchor is on a lined field with a "rope border" encircling the emblem. (Albert button #NA-15.)
Clean the dirt off the "Great Seal" US Army button (which has a raised rim), to see if it has "black finish" on its front. If so, it dates from 1910 to 1923. If there's no black finish, the button was made after 1923. If you can make out a backmark on it, we might be able to narrow down it's manufacturing date range.
The large brass disc which seems to have grey-ish corroded solder filling its back and its front showing a Greek god/goddess with a winged helmet appears to be a horse-bridle rosette.