I'd like to take a stab at it.
First, please accept this as an addendum to the original question. Even though it was not worded as such, I believe it is within the spirit of the problem:
<addendum>
There is ample friction between the wheels and the treadmil such that traction is never lost via the plane's acceleration. Sure you could strap a saturn V on the back of the plane and incinerate the wheels as the plane heads up to escape velocity but I don't think that would follow the premise of the original problem. If traction is lost, then of course the plane will be able to move independent of the treadmill and the plane can take off. In other words, there shall be no skid marks on the treadmill.
</addendum>
It seems the schism here is between those who feel the wheels have absolutely no effect on the planes movement and those who do. Those who claim that wheel speed will simply double as the plane moves forward on the treadmill are of the "freewheel" camp. Please consider this scenario:
Plane is idle on the treadmill, engines off. Start the treadmill at 50mph. There are two possible outcomes:
1) Plane remains stationary, wheels are spinning at 50mph.
2) Wheels remain stationary, plane moves at -50mph.
I'm of the belief that the plane will be moving backwards and the wheels will be still. The freewheelers would think otherwise, the plane is stationary, the wheels are moving.
Scenario 2:
Plane is reversed on the treadmill such that they are both moving in the same direction. Takeoff speed for the plane is 200mph. Plane is idle, engines off. Start the treadmill and accelerate it to 200mph.
A) Plane remains stationary, wheels are spinning at -200mph.
B) Wheels remain stationary, plane moves at 200mph, lifts off from runway and promptly lands on runway as it quickly loses its forward speed where it is again acclerated up to 200mph, repeat. (At 200mph air resistance will likely be enough to overcome the static friction between the wheels and the treadmill causing the plane to start rolling backwards so you might have to speed up the treadmill a bit to get the plane to 200. Or, the terminal velocity of the plane may be less than 200mph and it will never take off with the wheels accepting the excess velocity but I think the logic still applies.)
Again, I see scenario B to be the more likely outcome. The freewheelers would think the plane just sits there. Going back to the first scenario and assume that the plane was moving backwards. Accelerate the plane such that it gains on the treadmill until you reach the origin. Now, back off on the accelerator so that you are no longer moving with respect to the ground. Here, I see the plane at zero speed relative to ground, treadmill at 50mph, wheels spinning at -50mph. Now, speed up the treadmill to 100mph. I would think the plane would again start moving backwards. If you can visualize this, then I think you can see that the plane can never advance if the treadmill is matching the plane's speed.
If the plane is independent of its wheels and therefore independent of the treadmill, then I can see how you think the plane can takeoff. This would only be the case if the wheels and bearings are frictionless which I don't think they are. Or if the air resistance of a stopped/low speed plane is greater than the friction between the wheels and the treadmill which I would also doubt. For example, in the first scenario if instead of air, the plane was trying to move through honey, there might be enough resistance there such that the treadmill would spin the wheels instead of move the plane. Also, I believe that whole issue of wheel driven versus thrust driven does not apply here. Those would be more relevent in the case of a dynamometer. A car on a dyno would not move. A plane on a dyno would.
As for the skateboard fan, it would not move as the fan is pushing itself backwards as much as it's pushing the board forwards. This would be the same thing as attaching a huge magnet a foot in front of your car and expecting it to move forward.
The idea behind pulling a rope is not a proper analogy. If you are on skates or whatever on a treadmill holding yourself by a rope, the rope is pulling you at the same speed that the treadmill is pushing you. If the treadmill increases speed, the rope will pull you harder. As long as traction is maintained between the wheels and the treadmill, the push/pull will be equal resulting in no net movement. If you now start pulling on the rope, you are adding a force outside of the system than can not be offset by the treadmill as any increase in the treadmill speed will still be taken care of by the rope. If the treadmill is pushing at -10mph, the rope is pulling you at 10mph. If you pull the rope at 1mph, treadmill increases to 11mph, rope pulls at 11mph, you're still pulling at 1mph, so now you're going 12mph. This cycle will continue until the rope shreds your hand or the rope breaks. The same occurs with the hand pushing the wagon from behind. If you think that the wheels spinning plays no part in the overall movement of the object, than you shouldn't need a rope or a hand to keep a wheeled object stationary on a moving treadmill.
P.S. I'm quite impressed at the discussion thus far. It has been rather civil and pretty open-minded. Hope it keeps up.