A thing to remember is, that a plane on the runway does not gain speed by powering the wheels, it gains it by the engines forcing air through. If a plane was on a conveyor belt that matched the planes speed, as soon as the plane starts positively accelerating, the conveyor belt would have no effect on the plane as the wheels are simply free turning.
There are many variations of this. If the conveyor belt exerted more "friction" like force on the plane than the engines were producing thrust, the plane would continue to move backwards with the belt, until the force of the engines overbalanced the force of the conveyor belt, the planes -ve speed would then reduce and would increase in a positive direction.
Another thing to note is that a plane will only be able to take off when at a certain air speed. For example, lets say the plane can gain enough lift to take off when it reaches a speed of 240 knots. If the plane was in a headwind (wind coming towards the front) of 100 knots, then the plane would only need to accelerate to a ground speed of 140 knots before it could lift off. Likewise, if there was a tail wind of 60 knots, then the plane would have to accelerate to a ground speed of 300 knots in order to reach the desired air speed.
Just to clear up a few other things, when a plane is climbing, the engines are providing more of the lift than the wings as the aerofoil shape is inefficient at that angle of attack. The engine is providing the power to climb.
To get a planes nose to rise, there needs to be enough air flow of the tail of the plan, as this is where the elevators are located. To bank a plane, the tabs (ailerons) on the end trailing edge of the wing move accordingly and disrupt the flow of air. When a propeller plane is flying with its nose high, and with its throttle fully open it will be flying slowly, however it can raise and lower the nose fast as there is air flowing over the tail caused by the propeller. Its banking is sluggish, because there is little air flowing over the ends of the wings.
Sorry, I haven't checked what I've written, but if there's something you want to know, let me know.