The answer to both of your questions is: both of them die. This is because Deathtouch "works" during a fight, while First Strike does not. I realize this is a little counter-intuitive, but the way First Strike actually works is to create an additional damage step in the combat phase. Since the fight mechanic isn't the same thing as your combat phase, First Strike never triggers.
Think of it this way - Fight just makes both creatures do damage to each other. Period.