This'll be a long post (cause I love Physics)
Let's go back to the original hypothetical situation:
A lot of people have gotten hung up on the conveyorbelt motion vs the wheels motion debate.
Howver, the hypothetical scenario doesn't what kind of plane it is, or if it has wheels or pontoons.

In any event, the type of landing gear does not matter, because the landing gear is not reponsible for any propulsion in the airplane whatsoever. The landing gear only allows for the movement and steering of the plane on land/water.

Take your old Radio Flyer Wagon, for example. The wheels of your RFW allowed you to steer the wagon, but they did not make the wagon go - either gravity or someone(thing) pulling it created the motion. The wheel rotation just allows the wagon to move easily.

The conveyor belt speed vs wheel speed is a red herring - You could easily assume the plane is a pontoon style seaplane trying to take off using a water treadmill.

I gather from what is said in the hypothetical sscenario that the plane will remain in one place relative to the ground at any engine/plane speed because of the conveyor moving in the exact same speed in an oposite direction. That means at any Plane/conveyor belt speed, the difference in velocity between the airplane's wings and the ground will always be 0 mph.
So the Plane is moving, the runway conveyor is moving, and the assumption is that the body of the plane does not move relative to the ground.

And the relative motion between an airplanes wings and the ground is irrelavent to a plane achieving lift.

What we have to remember is that we are talking about a plane, in which the thrust created by the engine is created by the engine moving the air. The airplane wing in essense does not care what the relative speed of the wing is to the ground, but rather what the relative speed of the wing is to the air. That's what is going to generate the lift required for takeoff...

Now their are other factors for an airplane to takeoff, but since we will assume that the plane would have no problems taking off under normal conditions (the "treadmill" is not what I would call a normal condition)

Lift is generated by the difference in velocity between a solid object (the wing) and a fluid (air - a gas is a fluid). There must be motion between the object and the fluid: no motion, no lift. It makes no difference whether the object moves through a static fluid, or the fluid moves past a static solid object.

Now while the airplane appears to be staying in one place relative to the ground because of the "treadmill" under the plane, we know that air movement must be taking place. We know this because the motion of an airplane is created by the engine moving the air. Without any air movement, the plane and treadmill would remain stationary.

Keep in mind we aqre dealing with velocity (directional speed).
Adding 200mph east to 200mph west = 0 mph overall movement, even though the two moving objects are moving at 200 mph be themselves.

Because the airplan wings are affixed to the airplane, the wing and plane speed are the same.

Assuming the airspeed is 0mph...

So if the treadmill is is moving at 0mph, that means the the airplane must also be moving at 0mph, and that means the difference in velocity between the wings and air are both 0 mph - no lift possible

Now if the treadmill is moving at 200mph (west), that means the the plane must also be moving at 200 mph (east) - and while the difference in velocity between the Plane and treadmill remains 0 mph, the difference in velocity between the wing and air is 200 mph.

For the treadmill to reach speeds of 200mph, the plane has to be travelling 200mph (in the opposite direction) and while the ground is not moving in relation to the plane, the air is...since the air has to be moving for an airplane to move (engines) and thus for the treadmill to move. Thus this means the assumption that the plane remains stationary on the treadmill is false.

So yes, the Plane will be able to take off on the treadmill, assuming the plane could take off under normal conditons to begin with.

As for the bullets - both bullets (ignoring air resistance) would land at the same time. Both bullets are subject to the force of gravity (~9.8 m/s^2 - with slight variances based on locations: due to differing densities of the underlying crust, that the earth is not perfectly spherical, and the centripetal force from the motion of the earth's rotation on its axis varies at different latitudes.) So at the moment of being fired or dropped, the bullet begins to fall towards the earth with an acceleration of about 9.8 m/s^2.

The fastest speed of a fired bullet is ~1500m/s (~467.3ft/s) - bullet speeds depend on the type of bullet. The time it takes a bullet to travel 32.8 meters (10 feet) at 1500m/s (476.3ft/s) is 0.021867 seconds. In that time, the bullet will have fallen towards the earth approximately 0.002344 meters (0.000714 ft)...that is pretty much inisignificant to the human eye, which makes it appear as if the bullet is defying gravity.
If the bullets are fired and dropped at a hieght of 19.6 meters (5.97ft), they will both hit the ground at the exact same time (2 seconds later), but the dropped bullet will be directly below the spot where it was dropped, while the fired bullet will be found ~3000 meters away (914.36ft) maximum (again, ignoring air resistance)