Alternating possession from opening tip
 

A1 tips the ball in the opening tip off towards A2. A2 makes an attempt to grab the ball but knocks it out of bounds. Possession arrow should now be set to indicate that team B will get the next alternating possession. Correct?

Nope. B gets the ball for a throwin (OOB off of A2), A gets the arrow.
Rule 4-3

Thanks for the reference. I guess the refs 5 min discussion did end up with the correct rule implementation last night. Hard time convincing myself that the rule is logical.

What if A2 attempts to dribble the tipped ball instead of just securing it first? Sect 12 art 1 "a player is in control of the ball when he or she is dribbling a live ball"

If his dribble started before the ball went OOB, then that changes the ruling.

Would you have a problem with the rule if A1 (the jumper) simply tipped it OOB?

"attempts to dribble" is not a dribble -- same result as the OP.

Depends what is the definition between a dribble and an attempted dribble?:) In my book they would look the same.

Just worry about Player Control, or not Player Control. If a player had "attempted" to dribble, would you have granted his coach a time-out during this "attempt" to dribble.

Maybe you mean "interrupted dribble?"