Try out box2dweb, its the as3 version of box2d converted into JS.
The way they work is that you put bodies into the physics world, and then step the physics simulation - it will resolve collisions and things to give you the new position of your rigid bodies.
Let your logic entities put rigid bodies into the physics scene when they are added to the world, and remove them when they are removed from the world. When you update, update the physics system first, and then let your logic entities get their new positions from their rigid body or bodies.
This will all make sense once you have done it for real.
So basically the physics engines don't and shouldn't work with real game entities, but abstractions? I'll look into the API, because that would be great!