How the gravitational force between two objects depends on the distance between them? What happens to the gravitational force between two objects when the distance between them is doubled?

Gravitational force F between two objects is inversely proportional to the square of the distance between them, i.e.,

When the distance between two objects is doubled, the force becomes one-fourth of its original value.

No, the value of ‘G’ is independent on the medium present between the two bodies.

G is a universal constant.
So, its value on the moon = 6.67 x 10^–11 Nm²/kg².

The value of universal gravitational constant is 6.67 x 10^–11 Nm²/kg². 

There exists a gravitaional force between all objects in the universe. When we move our fingers, force of attraction changes when the distance between two bodies changes. 
Therefore, the position of the stars also changes.