Most hawks are very territorial so playing is not really common as far as I know. There are several species that will do "sky dancing" as they fly at each other in a mating ritual. I have seen a...
Because it was easier than calling it a falcon? "Falconers" use falcons to hunt. "Hawkers" use hawks (the birds) instead of falcons. A plasterer's pose -- with hawk -- is quite like a hawker's pose....