It would vary on the species. The most commonly kept is the Chukar, which is used for the purposes already described. There are also a number of mutations of Chukar. Close behind is the Hungarian (Grey), also used for the purposes described.
There are a number of species that are kept for simple avicultural enjoyment and some for conservation. Our partridge/francolin section at gbwf describes a few of the species seen in our aviaries. Perhaps the most sought after partridge for the aviary is the beautiful Roul-roul. They are delicate and require more care than the other species.
There are also several species that have never been established in captivity or have been allowed to die-out. There were several Francolin species as well as some of the tropical partridges in American aviaries up into the early 80s, but have since disappeared.
Yes. Partridges as used to release and hunt. You can make big bucks if you set up a place and let people come hunt them. Some places around me raise ringneck pheasants to release and hunt, so people from all over come and hunt them, and if you shoot a cock you pay $60.. per cock! WOWZA!