I suppose it is possible but a full grown chicken is a little heavy for a hawk. Usually they will just take pieces and come back if it is still there.
Depends on size of chicken vs. size of hawk. We have redtails in our neighborhood that will take off with bantams & Japanese, no problem. Ask me how I know... :'(
I don't know the breed you are speaking of but the highest death rate for free range chickens is the family dog.
Some breeds are specifically bred to be livestock guardians, and are very gentle with all livestock (including poultry) with some training. The training part is mostly teaching them to stay calm when the chickens approach them, not to be too playful, building on their innate territorial behavior. Great Pyrenees are popular, but other LGDs include Kuvasz, Akbash, Komondors, Anatolians, Maremmas. Sometimes herding type dogs can also be trained to work as LGDs, or vice versa--my Pyr will happily herd the neighbor's cattle. And dogs with little prey drive (many giant breeds) are usually fine once they get past the playful puppy stage. However, most terriers, retrievers, sight hounds or coon-type dogs will happily kill poultry when left to their own devices. Since lots of families have retrievers or terriers of some sort, yeah, the family dog can be problematic.