Just read you OP - as to batting out of order.
ART. 2 . . . After the first inning, the first batter in each inning shall be the player whose name follows that of the last batter who completed his time at bat in the preceding inning. PENALTIES: For batting out of order (Arts. 1 and 2): 1. A batter shall be called out, on appeal, when he fails to bat in his proper turn and another batter completes a time at bat in his place. 2. When an improper batter becomes a runner or is put out and the defensive team appeals to the umpire before the first legal or illegal pitch, or, play or attempted play, or prior to an intentional base on balls or before the infielders leave the diamond if a half-inning is ending, the umpire shall declare the proper batter out and return all runners to the base occupied at the time of the pitch. 3. When an improper batter becomes a runner or is put out and a legal pitch or illegal pitch has been delivered to the succeeding batter, or an intentional base on balls has occurred, or all infielders have left the diamond if a half inning is ending, and before an appeal is made, the improper batter becomes the proper batter and the results of his time at bat become legal. 4. When the proper batter is called out because he has failed to bat in turn, the next batter shall be the batter whose name follows that of the proper batter thus called out. 5. When an improper batter becomes a proper batter because no appeal is properly made as above, the next batter shall be the batter whose name follows that of such legalized improper batter. The instant an improper batter’s actions are legalized, the batting order picks up with the name following that of the legalized improper batter. NOTE: When several players bat out of order before discovery so that a player’s time at bat occurs while he is a runner, such player remains on base, but he is NOT out as a batter.
In layman's terms.
If a batter is discovered to be batting out of order and has not completed his at bat - the correct batter replaces the incorrect batter at the plate and assumes the count at the time.
If the incorrect batter completes his at bat and the defense appeals before a play is made to the next batter (pitch - illegal pitch - pick off attempt appeal of another runner - etc) the batter who was supposed to have batted is declared out and all runners return to the original base.
If the incorrect batter completes his at bat and a subsequent play is made - then he becomes the legal batter and the game continues with the next batter in the lineup.