Beastmode, I went about this in a scientific manner. Based on the exceptional disassembled photo posted by Jpyle, the only round spring in the entire parts kit is the guide rod spring. I am unable to tell the exact size of your small piece of spring, but it is either broken off the end of the original guide rod spring, or it a secondary buffer spring that sits inside of the primary guide rod spring. It is rare to see them snap completely off, but it has happened!
I believe MisterMcCool might be right that the washer is to protect the end of the guide rod from spring dig/damage.
“Malo periculosam, libertatem quam quietam servitutem." (I prefer the tumult of liberty to the quiet of servitude.). Thomas Jefferson
"All that is necessary for the triumph of evil is that good men do nothing."
-Edmund Burke, Loosely translated from Thoughts on the Cause of Present Discontents. (1770)