There is a ten pack of coyotes where I live that has been going around and decimating other peoples poultry flocks. For my LGD's I have a 5 pack, I always run at least three outside at night. They are "house dogs" that are required to work but not necessarily every night although they usually want to.
I am not sure how many coyotes one LGD can handle, it would depend on the dog and the coyotes, too many variable factors to even try to answer that. Personally I prefer to be over dogged than under dogged, the larger the defensive presence encourages predators to seek thier meals elsewhere. It also means less stress on the working dogs and I think it can help them be in better condition for if and when they do have to physically draw the line.
As far as dogs and birds, well that is all I currently use the dogs to protect, ducks, geese, and chickens. Yes you have to train them to be extra gentle when they are young, they don't always come preset. Some definitely take more patience than others. My four legged "problem child/wiggle ox" that acted very inappropriate towards the birds during his adolescence and was not trusted around them at all until he got his brains when he was about two and a half years old. Since then, he just turned five, he has turned into second in rank of the "I Love My Birdies" squad. He has appointed himself the primary guardian of hens that are brooding out chicks. Once he even tried to break up a fight between two ganders which he quickly realized was a mistake when they turned and latched onto him and started wing flogging him. He didn't try to hurt them them, he just tried to get them to turn him loose.
There is absolutely no way that I would be able to free range my birds where I live without dogs to protect them. I have witnessed the geese sounding off and alerting the dogs to something they need to bark at several times. I can also tell when a broody hen has chicks hatching out under her cause the dogs get extra guardy and barky about a day before the first pips. As far as predator losses and dog losses I don't have those. Also the "predator" that my dogs have had to get into it with the most are other peoples domestic dogs.