The first button is a "Great Seal" US Army button. Having a raised rim around its edge means it is from 1910 or later. In the World War One era those buttons had a "black finish" to keep the brass from exposing a hidden soldier's position to the enemy. That coloration was discontinued in 1923 and not resumed later. I can't be sure from your photo whether your button has any traces of "black finish" remaining on it or not.
The second button is an imitation of the US Navy's button made for use on civilian clothing.
The third item is a "Great Seal" US Army horseharness rosette, first issued about 1902 and still in use. Of course, most of those date from before World War Two, when the US Army stopped using horses for almost anything except Ceremonial purposes. One photo below shows your Great Seal rosette on the horsegear. The other photo shows the brass bar on its back for connecting it to the leather straps.