After 31.25 I don't even bother with animal husbandry. It always ends for me with hundreds of animals walking around my dining room and trampling grass, starving, fighting, etc. I just butcher every non pet migrants bring on the spot, but keep a few cats around, Even cats are an annoyance with all that dead vermin.
All you have to do is pasture them. Non-grazers only need a 1x1 space and they'll be happy, egg layers need nests too. Immature offspring can all be shoved in a cage until they grow up and are either slaughtered or bred. I slaughter any stray grazers the immigrants bring in, because many of them require far too much space to be worth it, though sheep can be worth it since they only need a 4x4 space preferably dug out in a soil layer somewhere, and they can produce milk and wool as well as meat. Also, junk pets like rabbits and cavies which I certainly don't want breeding in my fort get atom smashed.
I once put two animals, one dog or something, and one bird in a 1x1 watchtower, and they were fighting, so I don't think a 1x1 space will cut it. Besides once they have litter, I'll have to pasture that too, or put into a cage. It is just too much work to even bother. I kept some grazers for milk and wool, but they produced it slowly, and it is easier to order milk and wool from caravans, if I even wanted it. Egg layers are OKish, because I usually can pick up the eggs before they hatch, but I generally don't need the extra food. Bees are way below being rewarding for the effort it takes to have them.
I just find it easier to slaughter everything, and even with 2 or 3 butcheries I have trouble keeping up.
As for dropping animals to their deaths, I've heard somebody build an automatic breeder for elephants with pressure plates, but I only ever managed drop traps with goblins (and it killed some of my Dwarves too).