What is the ruling on keeping birds in general, and keeping crows in particular, and does the ruling differ for keeping different types of birds?

Raising birds is a matter of custom, and the default ruling for it is permissibility, except for what the Shariah has excluded. Raising a carrion-eating, harmful crow is forbidden, because it is one of the "fawasiq" (noxious creatures) that the Shariah commanded to kill. As for the rook (crop crow) that does not cause harm and does not eat foul things, raising it is permissible, and eating it is lawful.