When does a player establish front court status? If he is standing on the mid court line, is he front court or backcourt? Player A2, standing on the line, catches pass from A1 who is in the backcourt. A2 then dribbles once in the front court, then dribbles in the backcourt.

A player with the ball, establishes front court status when the ball and both feet are in the front court. The division (mid court) line is in the backcourt. If the player is standing on the mid court line, they still have backcourt status. If he is straddling the line, with one foot backcourt and the other frontcourt, he is backcourt. However, lift the backcourt foot and reset it down in the backcourt, violation! If A2, in your example, dribbles into frontcourt (three points, both feet and the ball) and then returns, backcourt violation. This rule is misunderstood by many fans and unfortunately many coaches. That is why we hear the calls for a backcourt violation, when none has occurred!

Not to nit-pick, but your first statement is not correct.

It's not 3 points for a player with the ball, just for the dribbler. You explained it correctly in the rest of your post, however.

ART. 6 . . . During a dribble from backcourt to frontcourt, the ball is in the frontcourt when the ball and both feet of the dribbler touch the court entirely in the frontcourt.

Reading 4-35 and 4-13 will give you clear understanding of this.