It's territorial marking; if you go wiping out their claims, of course they're going to re-stake them.
The "dead things as presents" behaviour is less certain, but the common theory these days seems to be that they think you suck at hunting, so they'll try to teach you. This would certainly explain the "not-quite-dead things as presents" aspect...
I've also seen a theory that if cat brings you dead things, it means the cat thinks that you're not a capable enough leader for the cat, and that the cat thinks you're not good enough to 'hunt' for your own food.
In short: Cat: "WORK AND BRING ME GOOD FOOD, YOU LAZY ASS MASTER!! OR ELSE I'LL TAKE OVER YOUR ROLE AS MASTER OF THE HOUSE!! Oh well, for now, it's LUNCHTIME!!" (then it brings you a dead mouse)
From what I've read, cat display that "hunting" behavior is 'cause domesticated cats lives in a psychological state of extended kitten-hood, and the bringing dead thing part is to prove their hunting skill toward their parents.