A1 is dribbling the ball up court and drops the ball (lets it bounce on the floor) while she fixes her ponytail. After she is done she picks the ball back and continues down the court. Would this be a double dribble?

Depends
 

If she kept the ball bouncing and continued without "picking the ball up", its legal. If she picked it up as its sounds like, double dribble.

Re: Depends
 

If in the process of "dropping the ball" or "picking the ball up" she at any point touched the ball with both hands, or allowed the ball to come to rest in one hand, then it's double dribble.

Double Dribble
 

How about ball comes to rest [no bounce just rolling or sitting] on floor and then she "pops" it with one hand to get it started bouncing again?

I have a tendency to say violation but don't have a specific reference.

She didn't pick the ball up, she just dropped it and let it bounce on the floor until she was done with her hair.