Regarding attaching pieces of cloth to trees or bushes near holy places, it is true that it is a custom. I've never heard, however, that it is to obtain "luck." I've always interpreted it as a way to leave a little bit of yourself behind before leaving a place of pilgrimage. I really don't know. It's not an official teaching of the Church that people are supposed to do that sort of thing. It's just really a custom or a "popular piety," as some people say. I heard somewhere that it is one of those customs that goes back to pre-Christian times, and that it has pagan origins, but I can't say for sure.
My friends and I do it whenever we go to St. Antony Coptic Monastery in the Mojave desert in California. Only we don't use cloth; we use dental floss.
Once when we were doing it, one of the monks saw us. He didn't say anything, but he gave us an odd look. I guess Copts don't have that custom.
Regarding the "sacrifice" of sheep, they are referring to something called "Madagh." I don't know exactly how to translate that word into English, but I would not use "sacrifice," as it has the wrong connotations.
Basically, it's a blessing service that is said over meat before it is eaten. You do Madagh on special occasions, like a feast day, wedding or baptism. It is lamb meat or mutton and it is not supposed to be spiced, except for salt which is specially blessed. I know someone who was at St. Nerses Seminary and sometimes they would do a Madagh and take the meat over to a homeless shelter.
The "animal sacrifice" accusation comes in because in the Old Country you couldn't just go to the local supermarket and buy your meat all neatly wrapped in cellophane. It came from an animal and you had to slaughter it before you could eat it. In the old days, and I think they still do it this way in Armenia, prayers are said over the sheep before it is slaughtered.
Folk customs have also developed which would probably turn off people from the West, such as brides and grooms stepping in the sheep's blood for good luck, etc. This is an example of a local village folk custom though, not something the Church tells people to do.
The priests themselves would not be actually slaughtering the animal, as I think priests are not supposed to ever kill anyone or anything, even an animal. I've heard more than once that there is a canon saying that a butcher cannot become a priest.
Protestants will sometimes refer to Madagh in their polemics, accusing us of still offering animal sacrifices because we don't believe Christ's sacrifice on the cross was enough, etc. None of that is true, however.
Madagh is briefly explained here:http://www.hyeetch.nareg.com.au/religion/feasts_p4.html#Madagh