What happens when we destroy the villain? Obviously, we have accomplished our "goal," but then what? Do we just take continue exploring until we find the next chamber? Does the death of the villain affect the other monsters spawned in the chamber (do they simply drop dead)?
No, the don't drop dead. You can choose to avoid them, but they will hunt you.
I'm thinking that having two chambers just adds an additional victory condition. In this case, destroy the villain from chamber one and Ashardalon from chamber two to win. But I'm not certain because the only specific victory condition listed in the adventure is "destroy Ashardalon." A strict reading would allow us to run away from the first chamber, seek out and destroy Ashardalon (I know, easier than it sounds) and end the game.
Your main goal is to destroy the Dragon. The first villain is just an obstacle, a "bodyguard", if you like. Yes, you can run away from the chamber and search for Ashardalon, but it's not recommended with a villain and some monsters after you (although it would be very fun).