O.K. The play: A1 takes two dribbles from the top of the key towards the wing...jumps (jumpshot) for a try and the basketball rockets out of his hands straight up in the air.

NOTE: A1 was in the air when this happened.

A1 returns to the floor and then catches the ball (first to touch).

Is this legal? If so are they allowed to pivot, dribble, shoot, pass, etc ...

Depends...
1. on whether this is an NCAA or NFHS game
2. on whom you ask

Go read the old thread. Tony posted a link to it. Within that one is another old thread on this. Read that one too. Then you can answer your own question.

How do we draw the line between a fumble and a shot that "came out of his hand funny?" If it has been established that A1 "went up for a shot," then who are we to critique the quality of his release?

Nobody really knows what the call is here?

I wouldn't say that. What I would say is there is no consensus on this play. Some know but disagree with others who know. :)

If that play happens in your game: what's your call?

Travel. The restrictions on what a player may do after lifting his pivot foot are explicit, and the rule doesn't allow for a player to lift his pivot foot and then fumble.

O.K. New play, same situation:

A1 comes to a stop after dribbling. Shot fakes, with both feet on the ground (feet never leave ground, i.e: he does not appear to be jumping for a try) ... the ball rockets out and straight upward.

Can A1 move and catch the ball and if so do what?

Pass, call time-out, or shoot. He gets to establish a new pivot foot. He does not get a new dribble.

Change it slightly. He takes one step with his non-pivot foot and lifts his pivot foot; then fumbles. I've got a travel for the same reason I do on the airborne fumble.

4.15.4 SITUATION D: While dribbling: (a) A1 bats the ball over the head of an opponent, runs around the opponent, bats the ball to the floor and continues to dribble; (b) the ball bounces away but A1 is able to get to it and continues to dribble; (c) the ball hits A1's foot and bounces away but A1 is able to overtake and pick it up; or (d) A1 fumbles the ball in ending the dribble so that A1 must run to recover it. RULING: Violation in (a), because the ball was touched twice by A1's hand(s) during a dribble, before it touched the floor. In (b), even though the dribble was interrupted it has not ended and A1 may continue the dribble. In (c), the dribble ended when A1 caught the ball; and it ended in (d) when it was fumbled. Even though the dribble has ended in (c) and (d), A1 may recover the ball. (9-5)

After recovering, the only thing this player may not do is dribble again. That would be a double dribble violation.

Adam,
Calling a travel here based upon 4-44-3a runs into the same argument as the airborne player situation of 4-44-3b. Namely, all of the prescribed limits on foot movement in 4-44 only apply to a player "while holding the ball." It says so right there in the first sentence.

In your play once the player fumbles he is no longer holding the ball. Therefore, the traveling doesn't apply.

I'm not sure I agree. The limits control what a player may do after lifting his pivot foot. Starting a dribble does not constitute "holding the ball," yet you call him for traveling if he starts it after raising the pivot foot.
Like I said, I'm still willing to think this through.

While it is true that that is traveling, it is not because of 4-44-3a, but rather part c which specifically states that action is illegal.

There is no such specific statement for a fumble.

If I were to apply the same logic you used here, "The restrictions on what a player may do after lifting his pivot foot are explicit,..." to 4-44-3c, then I would point out that this rule is also explicit and it only forbids starting a dribble after lifting the pivot foot. It does not say that the pivot foot may not be lifted before the ball is released when fumbled! :)