Running shoes are shoes exclusively for outdoor activities like jogging. While basketball shoes are shoes that can be used both indoor and outdoor activities like inside a basketball gym and...
Yes, but they are made for running in very rough terrain. For running in the park they are too heavy and bulky, and the soles are harder than regular running shoes, generally.
Airwair running shoes have a decent reputation. But the best way to determine if they are good for you is to go to a local retailer, try them, and see how they feel.