Can an offensive soccer player run up the field out of bounds?

Answer 1

Nothing. When the ball goes out over the sideline, there's a throw-in. If it goes out over the goal line (but not a goal), if the defending player or goalkeeper was the last one to touch it, it's a corner kick, and if it was an attacking player, it's a goal kick.

Answer 2

It's legal for a player to run off the field if it happens in "the normal course of play," such as while trying to play a ball near the lines or retrieving the ball from out of play for a restart. If a player leaves the field without the referee's permission and not in the normal course of play, that player should be cautioned (yellow card).

Answer 3

You cannot kick a soccer ball out of bounds and still play it. If the ball is kicked out of bounds you either do a throw in, corner kick, or goal kick, depending where you are on the field.