Your comments are all so familiar to me, as we have three cats. 
Our most proficient hunter is named Smudge, and he patrols around the poultry area and keeps it rodent free. He knows those rodents come up there for the grain! We let the chickens free range and often when we do, we see the chickens following him.
One day I kinda followed him, too, to see what made them enjoy his company. Turns out he is killing rodents, eating the parts he likes and then they finish the bodies off! The tail and head become fodder for chicken-football!
Geeerooooooooos! 
We only find bodies now around the front door where our other cat, Mr.Stubby,  hunts and the chickens can't go. My favorite though was when Stubby brought half a dead something and laid it on my pillow. Quite a way to wake up.
Thank goodness our third cat lives indoors......we already have the front and back yards covered!