A quarterback.
When a basketball player runs with the ball with out dribbling, they are traveling which is against the rules.
Traveling
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A player that runs out of bounds on purpose can not be the first player to make contact with the ball or the ball carrier. If they are pushed out of bounds by a blocker and make and effort to get back in quickly, they can make the tackle.
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The Quarterback(QB) or the Running Back(RB).
Receives the ball from the center, and either runs with it or passes it to another player. And they get tackled. A lot.
no
2 players cannot be made out with one ball. However, on occasions with "Timed-Out" there can be two outs after one ball.
That player is called the point guard.
Running! Although sometimes referred to as "rushing".