Okay lets create another analogy.
You place a free wheeling cart on a treadmill. A light shines from it onto a light sensor. If that light moves forward of the sensor the speed of the treadmill will increase, in the opposite direction. A GPS locator is placed on the cart, this will measure the velocity of the cart in relation to the earths surface, and in effect, through the air that surround sit.
As you push the car from the back it moves forward on the treadmill but the sensor senses this and increases the rate of the treadmill. The harder you push the cart the higher the speed of the treadmill. Compare this pushing force to the thrust of an engine.
The GPS locator shows no movement of the cart over the earths surface. If the cart does not move, the air does not flow past it. If there were wings there would be no lift.
The conveyor does not change velocity in proportion to the wheels, but to the force of the thrust.
[edit]Now let's get to the plane. The plane is propelled by its engines, "screwing" themselves through the air. [/quote]
Untill the plane reaches terminal velocity, the engines are more like pushing air backwards.
The wheels are there simply for the plane not to lay on the ground, resulting in a fairly low rolling resistance. Now, as the tires are not the devices that push the plane forward, rather than the engines, the thing pays little attention to what its tires do.
No. the tyres allow the plane to accerlerate with mimimum resistance. The plane will not reach take off velocity instantly. Again, that is why runways are so long.
The only result the conveyor belt will produce is that the tires will go twice as fast as the plane.
So when the pilot turns up the engines, they will start pushing the plane forward through the air, and the fact that the belt underneath matches the speed in the opposite direction has no negative effect on that. So if the plane does 100mph at takeoff, the only result is that its tires will do 200mph (100 by the plane and 100 by the belt). Still, this does not prevent the plane from taking off.
No, the tyres will rotate in the equal and opposite speed as the conveyor belt. The speed of incident between the tyres and conveyor will be exactly double that of the speed of one of them individually. However, when you add the speed of rotation of the conveyor belt to the speed of the wheels the resultant is ZERO. You seem to think the tyres are a seperate being to the plane. They are not. If the wheels have no resultant over the ground it is on, the plane will remain still, the wheels will rotate.