Clearing HotD 2 Original Mode unlocks the HotD2 Mini-Game. Upon selecting HotD 2 from the main menu next time you play, you will begin a HotD 3 based mini-game with prizes being items for HotD 2 Original Mode. It cannot be played again until you have clear Original Mode again.
In order to unlock the one-player only extreme mode for House of the Dead 3, you need to beat both House of the Dead 2 and House of the Dead 3. Extreme mode requires you to hit weak points to kill enemies and allows the use of a blocking attack.