There are at least two different ways to play high-low split poker: "declare" and "cards speak". With the "declare" option there's also "in-turn" and "simultaneous" (simultaneous declarations are done by players arranging chips in their hand under the table and exposing them at the same time). The exact rules should be determined before the game starts.
Generally in "declare" games, a player who declares low cannot win high (and vice versa), so player two in your game is out of luck. Also, a player who declares "swing" must win both high and low to get anything (i.e., he cannot win only high if his low is beaten).
Casinos rarely, if ever, play "declare" games. "Cards speak" is easier and leads to fewer arguments. Under these rules, all players simply show their hands and the best low and best high split the pot. To increase the chance for players to scoop a whole pot, "cards speak" games are generally played with a qualifier for low: that is, a player cannot win low unless his hand is, say, 8-high or below.
High-low declare games are NOT played with a qualifier, so if a game is announced as "Stud-8" or "Omaha-9", that implies cards speak with the given qualifier for low.
In all poker games, if the hand reaches showdown, any remaining player may ask to see any hand, but it is generally both a good idea and good etiquette to allow players to fold without showing.