I've never had any big problem with butchering.
I recently saw somewhere on the wiki that a workshop can butcher bodies within 43 tiles of the workshop. In addition to that, workshops can also butcher bodies stored in refuse piles anywhere (both fresh bodies of kills to yield meat etc, and old skeletons to yield bone). If you don't turn off auto butcher both of these are automatic.
I don't use hunting, but my understanding is that the hunter is supposed to haul the body to the butcher (or possibly to a refuse pile) to get it butchered. I think I've seen something about a bug with hunter hauling, though.
As far as I know, the horror of death generated by dead bodies applies only to bodies of sapients, although some dwarves might be over sensitive and care about animal kills as well. You should keep bodies (and parts and teeth) of sapients away from the regular refuse pile, away from dwarven eyes, and away from merchant eyes, since they cause bad thoughts every season they're see (and in the case of merchants, causes them to bolt and leave without any other indication than an exclamation point above the head of the merchant that saw the offending piece of invader. Bodies of sapients cannot be butchered, but if bones become available through "natural" butchering (such as severed arms from an axe) those bones can be used for bone based production, except for dwarven bone.
Butchering tame animals in cages is the same as butchery of any other tame animal, while wild caged creatures have to be killed in combat. I think you can dedesignate the creature from the cage to have it released (and possibly hurt the civilian releasing it), but the safer method is to build the cage and then hook it up to a lever and have a civilian pull the lever while the cage is surrounded by a militia squad.
It can be noted that the default setting is not to collect bodies from the outside, so any bodies of combat killed creatures on the surface will not be collected unless you change the 'o'rders-'r'efuse settings.
Also note that combat killed tame creatures cannot be butchered at all, so when the goblins kill a draft yak it's lost (I think getting a necromancer to revive it and then kill it again will make it butcherable, and I also think that's a way to revive rotten meat, since I believe I've seen creatures start to create miasma both after the original kill and after being killed while reanimated). Tame creatures can only be butchered while still alive.