Great game idea, but this is a 0.3, not a release version. It's only fun if you're experienced and playing with a group of people on the ball. Otherwise, it's a drag.
Figured out the level requirement thing. Turns out your character level is persistent across games.
To get to the end of Adventure mode, you have to build up your experience points in Arcade mode. Once you've beaten a level, you can always start the following one, so quitting adventure mode and going back to the menu doesn't hurt you.
This is a perfect little game. Funny, friendly, enjoyable, stress reducing, and hugely satisfying. It makes me wish I could favorite developers, so I know when more magic comes out.