And a plane so you can fly off to cooler climates.
Well if you have so much property that your lake is 9 miles from your house, you can surely afford gardners to mow around the lake.
Either the cat likes you or doesn't want to feel indebted for the food 
So do our house cats. Not the most fun thing to find. DW found a dead bat in a doorway yesterday morning. Not sure how it got in but I suspect the cats made sure it stopped flying. I guess flying mice aren't as tasty as terrestrial mice, there was no obvious damage other than a small bit of blood on the floor.
Clearly you have been paying attention to the behavioral aspects of your chickens.