Are we sure that a rubber band isn't a spring? I guess the instructor just doesn't want you to load a spring and fling something up in the air with it.

Any way I think that twisting the rubber band is the best way to go. You could just buy a bunch of rubber band powered air planes, and use the parts to make your helicopter. You'd definitely have enough rubber bands to get the thing pretty hgh!
Now then, back to rockets, which will go way higher

Here's a site that has a pretty sinple setup.
http://www.antigravityresearch.com/
Here's a vid of their rocket car. You can see in one of the clips just how fast it goes. I'm sure you can get one to fly quite high
http://www.antigravityresearch.com/HTMLobj-872/Rocket_Car_384kbps.wmv
I'm sure that you can get a rocket up without a launch pad. I'm sure it'll take a few test fires, and some fine tuning to get the rocket to fly straight without a pad, but it is totally possible. I have launched large estes powered rockets that were only going 2 or 3 miles per hour when they cleared the launch pad, and they flew nice and straight

Take a look at the arial photo they took with one of their rockets. It looks like it was taken from a good 1,000 feet. You can totally smoke your class mates with somehting like this.