Not in the NBA- that is called a travel.
In the NCAA, yes, provided it is a bonefide shot attempt, in the opinion of the official. Rule 9, paragraph 6 of the NCAA Rule Book indicates that any shot attempt constitutes a loose ball and hence, anyone can recover it.
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No, they can't. They can hit the backboard or the rim and rebound it, but if the shot does not hit either of them, then it is considered a travel, or as others call it, a "self pass."
Yeah, if you can reach it in time before it goes out of bounds, or before someone on the other team dose.
Yes but only if you catch the ball in the air and shoot it while you're still in the air, otherwise traveling will be called.
If it is a shot, yes. Even if it hits the floor it is not a travel. You can always rebound a missed shot attempt, even if it‘s an airball and it‘s your own miss.