1. It's a violation, unless the player advances their feet into the front court as well. Rationale (unofficial, but sensible): the first dribble established front court status; the player must advance with the ball.
2. This situation depends on the following circumstances (note - the player cannot establish both front and back court status):
a) if both feet were on the floor when the ball was caught, the player may advance the backcourt foot to the front court (no violation, unless they travel), but may not bring the frontcourt foot to the backcourt or dribble in the backcourt (violation).
b) if the player catches the ball with one foot in the backcourt and then puts the other foot down in the frontcourt, play on.
c) if the player catches the ball with one foot in the frontcourt, they cannot put the other foot down in the backcourt or dribble the ball in the backcourt (both violations).
In the situation you describe, as long the player's next movement (step, dribble or pass) is in the frontcourt, there is no violation. If their next movement is to the backcourt, they have committed a violation.
