The waves consist of lots of wave movieclips that animate with shape tweens. They are given random x coordinates every time they loop, and they all start off on a different frame of their animation.
The fire, if I recall correctly, works the same way, except instead of being given random x coordinates, they are given slighty random rotation, width, and height.
The bouncy hide-out thing is strictly animation. Shape tweens for the ropes and motion tweens for the other things. The upper half is one animation, and then the bottom half is all inside the movieclip of the little square thingy with a second animation inside it.
As for the bouncyness a lot of it is probably thanks to using full easing on all tweens. (100, -100)