The player gets under the flyball and it hits his glove and then starts to bobble around and comes out  -- is the batter out from the flyball?

The answer is no whether it's a throw, flyball, line drive, to be considered a catch it must be thoroughly controlled and the next action that would follow be done. Example flyball catch and then a step toward throwing it--bringing the ball out of the glove hand always constitutes control and the loss is not out of the glove but out of the bare hand.

Control is always the umpire's decision in the defensive player secures and controls the ball then that part of the play is over and moves on to the next part.

You can't bubble it as a fielder.