As I understand it, it is pork that needs to be bled, hence why it is a paler meat than beef or lamb. And I was told, by a man who did home kills on farms in Wales, that you stun them and string them up, then slit the throat so that the heart is still pumping to push the blood out. The pig dies from lack of blood carrying oxygen to the brain while it is unconscious from the stunning. I did in fact witness such a kill so that I would know what was involved and whether I thought the pig knew anything about it, and with this slaughterman I fet very comfortable that the pig knew nothing and was not stressed for one second at any time throughout.
The rules about home kill differ from country to country and species to species and are VERY strict.
As a commercial sheep farmer I am pretty sure that, in England at least, any dead sheep has to be notified and disposed of by an authorised deadstock handler. Our local hunt is authorised, as are a couple of 'dead cart' operations. All three will terminate a sick animal before loading if required at no extra cost.