The universal law of gravitation (Newton's) is: Force F of Gravitational attraction between two bodies of masses M1 and M2, separated by a distance d
F = G M1 M2 / d², where G = Universal gravitational constant.
The direction of force is along the line joining the centers of masses of two objects.
The Moon attracts the Earth with a force F and the Earth attracts the Moon with the same force F. The reason is the above formula. Another reason, is Newton's third law. Another way to understand is when molecules and particles on Earth interact with molecules and particles on the Moon the interaction is felt on both sides. So they experience the same net effect. It is the symmetry of position and matter.
2) The law is useful in finding the mass of Earth, and Moon. It is useful to find the gravity on a planet. It is useful to launch missiles, rockets and satellites. We are living on the surface of Earth and water in oceans stays on Earth due to this law. We can calculate the planet distances, orbits and time periods (years) using this law.
3) The gravitational force exerted by the Earth on an object is called the weight of that object on the Earth.