Why doesn't the moon fall down on Earth?

Actually it is constantly "falling". Orbit is constant free fall. What keeps it up is the force acting contrary to gravity. In this case, the momentum carrying it forward in its orbit is trying to carry it away from the Earth and gravity keeps pulling it back.

These forces must be equal or the distance between two bodies will change. In the case of the moon, its orbital momentum was created when a Mars sized body plowed into the Earth. Gravitational pull is related to the masses of Earth and Moon and their distances from each other. In our case, they aren't equal. As the moon loses momentum, its speed drops and its orbit expands. Eventually, loss of momentum will eventually make the moon tidally locked over one point of the Earth';s surface, taking about 47 days to complete one orbit.

Moon is pulling the earth and the earth is pulling the moon, with equal magnitude in force but opposite direction.

Is more like when you swing a with a string to revolve around you, you are pulling the ball but the ball never fall to you, instead it revolve.

The moon stays pretty well where it is because gravitational attraction between earth and the moon is balanced by the centripetal acceleration of the moon circling the earth. Likewise, the earth + moon circle the sun.
Actually, the moon is very slowly moving away from earth. At one time, it was part OF earth.
Just where do you think 'down' is, in space? There's no up, no down, no left, no right. If there were a 'down', and assuming that would be south, the only habitable place on earth would be at the north pole region. Everybody else would fall off!
I can assure you, we living in the southern hemisphere are just as glued to earth as anyone in the northern hemisphere.