Young Doggett is shown imbibing a bottle of Mt. Dew prior to going to the Social Services office. The contents are obviously water, as Mt. Dew is a green, cloudy beverage, noticeable even though the bottle itself is green.
It makes no sense that the tunnel was not discovered. After Vee used it to break out, there would have been a search, and since Caputo came in as new assistant warden, the search would have been extra thorough.
Contrary to what Luschek says, if you rent a gun on a shooting range, you need to pay a deposit before they hand it to you.