Re: train them to target vermin?
My cats have bought back pigeons, frogs, assorted mice and voles, several rats, a budgie and a slow worm(!). Almost always alive. Catching the buggers at 3am is no fun (especially the rats). I can only think they see me as a crap hunter that needs more practice. Especially at 3am.
Releasing a pigeon from a box in your front garden just as a curious drunk walks past gets a great reaction though ;-)
Now one's dead and the other is elderly, so mostly I get brought leaves now (really!). I guess it's the thought that counts.