That would work but if you do the owner will take a bad thought hit thats worse than the miasma. Keep your cat population under control by caging new born kittens. You can trade them away to caravans or whatever, personally I keep them all in a cage as you never know when more vermin might show up, releasing the odd cat is a good idea in that case. If you put new cats in cages they will not adopt owners and you can do what you like with them, they will also not breed while in the cage - one thing I recommend doing with cats, indeed any animal you want to control the population of - is to keep all the females in one cage and all the males in another, when you need an animal for something other than slaughtering (guard dogs, mouser cats etc.)
only release male animals in whatever quantity you need*. You will no doubt have at start some female cats who are pets and can't be caged, when this pet eventually dies make sure its the last female thats ever out of a cage - except for in a circumstance where you want to breed more, in which case release a single female into an isolated room with a breeding male, then recage them after the new ones are born, along with any new ones you don't have an immediate use for. You're quite right about the cats killing vermin and taking it to their owner, the owner often ignores this and its up to other dorfs to dispose of the corpse - what I do is designate a spare peasant or migrant as a 'Refuse Collector' his only job should be to collect refuse, in a larger fort more would be needed too.
As a cat owner in real life I can never bring myself to kill them in DF, silly I know. I can be very mean to dorfs, goblins, ratmen etc. though
- as my 'This is Sparta' lava pit and bear pit will testify.
* interestingly this makes sense from a realism perspective too as in real life bulls get slaughtered for beef except for a stud for breeding - the cows make milk and produce calves, likewise male dogs are more often used for guard duty/warfare as they are more aggressive etc.
edit: on the subject of breeding animals I just had the weirdest thing happen - I have a female horse and a male donkey, all of a sudden my horse gives birth to a horse foal, this is odd, I have no other horses, I check the units screen and there are no other horses on the map (i know that animals don't have to be near each other to breed). Could the donkey have bred with the horse? If so why is it not a mule instead of a horse foal?