Rolling ball vs Dribbling
 

I have searched everywhere for a rule or a case that accounts for a player that rolls the ball, then picks it up and starts to dribble. Is this a violation? and if so, where is it covered in the rules?

To better understand what you're asking, why would a player roll the ball, pick it up and start dribble? Have you seen this done?

Actually yes. During a scramble for a rebound, the ball is laying on the floor. Player A pushes the ball, rolling it away from the group on the floor, then picks it up and proceeds to dribble the ball away. That got me thinking of all types of other similar circumstances that could come up and looked for a rule about it, but found nothing.

If the player just pushes the ball and never has control, that player is free to recover the ball, pick it up and start a dribble.

This players hand never left the ball. They didn't just swat it away. Imagine the player putting their hand behind the ball and pulling it away from the scrum on the floor. The ball is rolling on the floor with the player guiding away. They then stop the ball, by pinning it to the floor, and then pick it up. That seems like a controlled effort.

By definition, a player is in control when holding or dribbling the ball, so pushing or dragging the ball across the floor would not be considered control.

So with that definition, and I know this sounds silly, but that would mean a player could keep batting the ball around the floor, rather than dribbling, as long as they never grasp the ball or push it to the floor to start a dribble?

Yes they could....now, why 5 defenders would allow this to happen is beyond me.

As BadNewsRef said, yes, he could. Is it realistic? No.

A player could also tap a rebound and continue tapping it all the way down the floor. Is it realistic that that could happen? No,

Player control is not established until the player is holding or dribbling the ball.

We get to adjudge the start of the dribble. ;)