Whether or not a plane will lift off or not has little to do with the ground or belt beneath it. It has everything to do with the air rushing around it. Normally, when a stationary plane needs to gain this "airspeed" it involves traveling on the ground a bit.
Imagine, in the vacuum of space, that you have a plane either on a treadmill, or using rocket burners, or whatever sort of locomotion it has. If there are no air molecules pushing around the plane, it will not take off.
So, in response to the conveyor belt question:
You said the belt is going AGAINST the plane, but traveling at the same speed that the pane is moving itself. I take this to mean that the plane is actually stationary (like a person on a treadmill, or exercise bike...not really going anywhere). No, it will not take off, unless somehow, the belt itself is causing enough air movement towards the plane, which I find unlikely (<--a guess, without doing any experimentation).
Give thanks to Mother Earth for Her greatest gift...
For without it, we would be lost.