Your image in your signature leads me to think of something funny. According to the extremist Christians, God causes hurricanes, tornadoes, earthquakes, tsunamis, etc etc because he hates [insert group]. However, it's a proven fact that obese people die earlier than other people. Why aren't they saying that God hates fat people?
An airplane moves because it pushes itself through the air. The treadmill is connected to the ground, not the air. That means the plane would push itself forward regardless of the speed of the treadmill, right? I think I'm getting it.
The riddle in your sig stumps me. I always assumed that planes use the lift generated by their wing shape to gain altitude, but if the plane is staying in place so there's no wind, there would be no lift, right? Unless the thrusters are generating lift, like on a harrier? Or unless there's sufficient outside wind?
The engines don't send power to the wheels, so the treadmill has no impact on the plane's ability to take off. The engines are sending power to the propeller or whatever device is being used to generate forward momentum, which is then used to achieve take off. The plane will still move forward and achieve flight normally because the engines/propellers are pushing the plane forward, not the wheels. The wheels do nothing for the plane, other than give it a means to make contact with the surface while still being able to maintain the hull's integrity (in other words: without crashing).