Tennessee Digger is right... button #1 is the 1875-1902 version of US Army "general service" uniform button for Enlisted-men. Military button collectors call that version the Indian Wars eagle-button... and have nicknamed it the "chicken-eagle" button because the eagle's head, neck, and short wings look somewhat more like a chicken than an eagle.
Button #2 is a civilian usage brass "fancy" flatbutton. I think I can discern enough of its backmark's lettering to tell it says "Benedict & Burnham." That company was in business with that name from 1834 to 1843.
Final photo:
The object in the center is a civilian-usage horseharness rosette, from the early-through-mid-1800s. All of the buttons surrounding it appear to be brass 1-piece flatbuttons, dating from the latter-1700s through about 1840.