Regarding the question about who or what exerts the force called the "weight of the object" on an object near the surface of the earth:

The correct choice is "The earth." Another correct answer is not provided in the list of possible answers. Regarding the gravitational force, there are two points of view:

The first point of view is called the action-at-a-distance point of view. In this point of view, the earth directly exerts a gravitaional force on the object without touching the object.

The second point of view is the field point of view. In the field point of view, the earth creates a gravitational field everywhere around itself. The field is a force-per-mass associated with every point in space in the vicinity of the earth. Near the surface of the earth the strength of the field is 9.80 Newtons/kilogram. It has no effect unless there is an object in the field. The field's effect on an object within itself is to exert a force on that object, directed toward the center of the earth, equal in magnitude to the product of the earth's gravitational field strength at the location of the object, and, the mass of the object.

The two correct answers to the question are thus: "the earth" and "the gravitational field of the earth." Only the former appears in the list, hence, it represents the only correct answer in the list of possible answers provided to you.