Yes, that should work.
Falconers who hunt with hawks and ferrets house the ferrets right next to the hawks. The ferret is sent down the rabbit hole to chase out the rabbit, and the hawk will take the rabbit and never take the ferret by mistake.
Hawks don't kill for sport, so a well fed hawk isn't going to kill any chickens, whether he was raised with them or not.
A hawk is a very small bird, so it is not going to do any protecting from anything. Maybe, if it is territorial, it might try to run off another hawk of the same breed and gender as itself. But maybe it would be driven of, itself, because a captive bird would not be as strong as a wild bird.
That murder of crows might drive away hawks, but it will take over the job of killing chicks and they will steal all the eggs they can reach. The best protection for your poultry is an adequate pen.