Is this traveling?
 

A1 is dribbling hard down court when he leaves his feet and passes the ball. However, his teammate did not see the ball coming. A1, realizing that the defense was going to get to the ball before his teammate does,catches up to the ball and taps it to a different teammate.

double dribble
 

I would call it a double dribble...dribble ended then the tap is a new dribble..

I have a travel. Batting it to a teammate is a pass. So his pivot foot came back down and after that he cannot pass it.

Before passing the ball or as part of passing the ball, did A1 end their dribble? If so, double dribble , if not, no violation.

Sam has the right idea.
You need to tell us if A1 ended his dribble when he left his feet and attempted to pass the ball. If not, then this action is legal.
Also, please tell us if the pass bounced on the floor prior to A1 getting to it.
If A1 did end the dribble, you could call a double dribble violation (listed as illegal dribble in 9-5) if the pass struck the floor since it meets the definition of a dribble, or a traveling violation would also be correct due to 4-44-4b "Neither foot may be lifted before the ball is released, to start a dribble."
If the pass didn't strike the floor, then it is an illegal dribble per 4.15.4 Sit D&E in the Case Book.

But I can't tell you for sure without more information.

Just to see if my thinking is correct, and a (slightly) different situation. A1, dribbling downcourt, goes to pass the ball upcourt, by pushing the ball up to A2, with one hand. A1's hand was on the top of the ball when he pushed it. After the initial push, A1 realizes that it will not get to A2, and runs up to get it, and does by dribbling the ball again, but the ball has bounced 2+ times. I think this would be legal, but am I correct?

Thanks.

To travel, one has to be holding the ball. How is this traveling? :cool:

You are correct.

I would say, no travel.

I was thinking it would be that the bat of the ball to a teammate could be viewed as regaining control of the ball. Which of course by definition it isn't. So I guess my thinking was some sort of much advanced, JMO, beyond everybody's comprehension, OldSchool, type thing. That's what I get for thinking.