In baseball what is the extra line-box going down the first base line and what is the ruling for a batter-runner going to first base Does he have to stay within that box?

Answer 1

That extra line creates a "running box". When a batter puts the ball in play, while running to first base he must run in that box until reaching first. If not, he may be called for interferrence should a fielder, usually the catcher, throw to first and hit the batter/runner if he is outside the box in fair or foul territory.