Yeap, I meant the latter, the conveyor belt moves away at the same speed at every condition. Meaning the plane is always stationary.

im in 'yes' camp..

y i said so? it because the plane use engine thrust (air) to create a lift. it doesnt depends on the rotation of the wheel. this means the wheel just only a tool to supporting the plane to move..so when the conveyor belt moving in opposite direction of the plane at the same speed (theoretically) while the plane is in motion, the only thing different is just the wheel is rotate twice as much compared when the plane take off on normal ground.

or simply, the plane use thrust created by the engine to move forward, not by the rotation of the wheel.

It's an empirical equation, so yeah at best we take it as some approximation (which will probably fail at some limit). Actually if you used the same linear approximation, given powerful enough engines, you would reach the same conclusion that a plane on wheels with brakes applied will also fly off given the same condition of a conveyor belt.

Also, the above explaination only explains the first part of my post, where I claim that in the ideal case with friction and no rolling resistance, the plane simply doesn't move when the conveyor belt moves.

The question is that can a plane take off on a conveyor belt moving at the same speed in opposite direction?

No, because the treadmill analogy fails to compensate for the fact that we're no longer talking about cuboid like objects, which experience massive friction that literally causes us to stop, but wheels instead, where the friction instead speeds up the wheel's velocity. Try this question then. Take a roller skater moving at say 5km/h. Let him skate to a walkalator moving in the opposite direction, at also 5km/h. Guess what will happen when he reaches the walkalator? [You could also substitute the walkalator with a treadmill, as long it's perfectly horizontal and stuff]

Solution in the form of an edit, after you reply (will take a while though)

Edit: 2nd analogy I just thought of: (True story)

When I was a kid, I always wanted to do roller skating/roller blading, and spent quite some time admiring and looking at my neighbour and cousin's skates. Naturally with my itchy hands I liked to roll the wheels using my hands by running my palm across the wheels. I hope you've done this too, but I noticed that the faster you move your hand across the wheels, the faster they rotate. Why? Same principle applies to the walkalator question, and same principle applies to the 'free spinning wheels' on the plane

do plane really need a wheel to move forward??
i mean, let say the body of the plane is on ice, the thrust from engine is enough to make the plane go forward.