Double Dribble or Travelling Scenario?

Q1: When a player picks up the ball after dribbling, bounce passes to a teammate but the pass is off. No one else has touched the ball during the process. So he goes over to pick it up, would it be consider a travel?

A1: No, this would be a double dribble. According to Article 24.1.2 A dribble ends when the player touches the ball with both hands simultaneously or permits the ball to come to rest in one or both hands.

Q2: When a player drives to the basket, picks up the ball to start a layup, loses the ball without anyone else touching it, can he legally pick up the ball?

A2: Yes, the player may legally pick-up the ball. To build on that, Article 24.2. A player shall not dribble for a second time after his first dribble has ended unless between the 2 dribbles he has lost control of a live ball on the playing court because of: A pass or fumble that has touched or been touched by another player.