As a general rule, airguns are allowed anywhere in the state of NY where shotguns are allowed. If it is stated that hunting is by bow only, then airguns are not allowed.
Westchester and Long Island are both bow hunting only, for deer. Small game is still allowed in Westchester with shotgun and airgun in some areas. Long Island is famous for the waterfowl hunting, so I can't imagine it not being able to hunt small game with a shotgun. Just be sure the land you hunt allows for shotguns/airguns.
By definition, and air rifle is not a rifle at all, since it does not use a cartridge. The definition of a rifle is to use a cartridge and a rifled barrel.
Good luck with the Nitro Venom, it's a great shooter for me up to 50 yards.