If our perception of space-time curvature is gravitation and Reduced Gravity Plane can reach weightlessness on some point of its trajectory, doesn't that mean that when Reduced Gravity plane reaches weightlessness? It actually bends space-time curvature BACK to what it is near the orbit. That is why Reduced Gravity Plane's crew experiences zero gravity.
It is a common misconception that we "feel" gravity. We don't feel gravity, we feel the "support force" that prevents gravity from accelerating us toward the center of the earth.
Normally gravity curves spacetime as you suggest, but we fight against that 'curve' by standing, sitting, etc. When the plane hits weightlessness, it is in freefall, meaning it isn't changing the curve of spacetime, but following exactly the curve created by gravity so that it appears spacetime is "straight" to the observer.
No. Feeling the "force" of gravity is not just an effect of the curvature of spacetime. The structure of spacetime -- whether it be curved or flat or whatever -- determines which paths through spacetime are "natural." These geodesics are the paths that, if you follow one, correspond to being in free fall.
You only feel gravity if you deviate from a geodesic. Standing still on the surface of the Earth is not a geodesic, so you feel gravity. Moving in a parabolic trajectory, as certain planes do and as you do when you jump in the air, is following a geodesic, so you feel weightless.
The plane doesn't fight spacetime. In fact, fighting spacetime and trying to forge your own non-geodesic path is precisely what causes you to feel gravity.
No, weightlessness is felt by any object in a free-fall trajectory in curved spacetime (technically, a geodesic path through the spacetime). You can read this article on the equivalence principle to learn more, but basically, the equivalence principle says that measurements of a freely-falling experimenter in curved spacetime should be equivalent to those of an experimenter moving inertially in flat spacetime far from any source of gravity. It also says that the measurements of an experimenter who isn't free-falling (like one at rest in a gravitational field) should be equivalent to those of an experimenter accelerating in flat spacetime (for example, if you were on a ship accelerating at 1G in deep space, this would create a feeling of gravity inside the ship).