The weight of a free falling object

Let's put things into more precise terms: we define "feel" as "there is a contact force between the two surfaces". Using this definition, it becomes clear that the hill does not "feel" the ball - if it did, then because there is a force, the ball accelerates (by $F = ma$), and its trajectory changes.

Regarding:

The counter-argument is, in the hill's frame, gravity exists, the ball accelerates (down), hence the surface feels the ball's weight.

If the surface feels the ball's weight, then by Newton's third law, the ball also feels the surface's reaction force - and therefore its trajectory changes. Since the question gives the ball's trajectory does not change, the only possible conclusion is that the surface does not feel the ball's weight.

Does it feel the ball's weight? Common sense physics says no.

What do you even mean by the hill "feel[ing] the ball's weight?"

There is no interaction between the hill and the ball; there is neither an exchange of momentum nor of energy between the two of them. In Newtonian terms, there is no force acting between the hill and the ball.

The counter-argument is, in the hill's frame, gravity exists, the ball accelerates (down), hence the surface feels the ball's weight.

In Newtonian mechanics, there is gravity, but the ball is accelerating downwards with the acceleration due to gravity. This means that there is no force from the hill pushing up on the ball. Why would the hill's "surface feels the ball's weight"? Your assertion makes no sense. If it is being pushed upwards, then it will NOT accelerate in the expected way. It might even stop moving downwards entirely.

In the General Theory of Relativity, gravitational interaction is due to the curvature of spacetime, and it is not a force like those in Newtonian mechanics. The ball's trajectory would be free fall if no force acts upon it; any deviation from the parabolic path is the evidence of a force, that we could identify as the normal reaction force from the hill pushing on the ball to make it not move into the hill, and maybe also a friction force. But you just said that the ball moved along the parabolic path that it was moving in the absence of the hill. This means that the hill being there is just not important and doing nothing.

An object in free fall is inertial and weightless.

You are getting confused on what "weightless" means. When standing upright, the sensation of "weight" that our bodies can feel is the compression in our spines that the bottom half of our bodies have to push up to hold up the top half of our bodies. If you hang on a pull-up bar, that sensation is reversed. When you let go of the pull-up bar and fall, for the short time of falling, your spine is neither compressed nor pulled apart, and that is the sensation that is called "weightless".

Even when you are falling, the gravitational interaction between Earth and you still exists. It is still pulling at you. In Newtonian mechanics, you still have a force called weight acting on you, just that you cannot feel the sensation that it usually makes you feel.