Does the fox hunt game exist professionally?
I've never heard of a fox hunt race in particular, but there is an extensive racing culture among bicycle couriers. These races are usually alleycat races, and generally involve riding to several checkpoints in sequence, sometimes performing tasks at those checkpoints.

Searching YouTube for bicycle messenger races, alleycat racing, and similar terms, will bring up many videos.

See also In Race of Bike Messengers, Potholes and Bad Manners Can Be Costly, a New York Times feature about the North American Cycle Courier Championship, a rare sanctioned race for which the city closed streets.