Bonnie, as I indicated initially I am a hobbyist, only. Take all that I tell you only as suggestion until you confirm it independently.
Whether the insignia for 1st Army was olive drab with a black A, or in color (as found on the web page I gave you), would be determined by the uniform on which it was found. It is the same insignia, signifying the same unit, in either case.
The red diamond on his left sleeve, I'm sure, would be the insignia for the 5th Infantry Division. In your first post you said your father was, " with Co.B, 2nd infantry 'Red Diamond'". This would suggest to me that he was in Company B, 2nd Infantry Regiment, Fifth Infantry Division. For some history on the 5th ID, see this web page: http://www.societyofthefifthdivision.com/WWII/WW-II.htmhttp://www.societyofthefifthdivision.com/WWII/WW-II.htm
You will see in this history that the division was, at different times, part of both First Army and Third Army. I would guess that your father was in 5th ID, and the division was for a time part of First Army (and that is when he came by the patches apropos to First Army and XXI Corps) and then the division was reassigned to Third Army, under Gen Patton. It's likely that your dad was simply one of the 20,000 plus troops of the division.
The "Ruptured Duck" is, indeed, an eagle, Bonnie. Everybody who was honorably discharged got one. Because it didn't have much significance, it earned no small amount of disdain among the troops, and became commonly referred to by the disparaging term: "The Ruptured Duck".
At that societyofthefifithdivision site which includes the history above there is more info about the division. There is other info available from sources online, such as this web page: http://www.army.mil/cmh-pg/documents/ETO-OB/5id-eto.htmhttp://www.army.mil/cmh-pg/documents/ETO-OB/5id-eto.htm
The division saw a lot of the war, and your dad's service is something you can be very proud of.