The positive sign must be taken for k to be real. Taking the negative sign would result in a negative value and hence k would be imaginary since the square root of a negative number is imaginary.
That "something" we see, we simply label it as a force that we call gravity. This "thing" we call a force, with the name gravity, has certain properties and it explains a large amount of phenomenon that we see around us. Otherwise, we'll be going into an argument about semantics if I'm...
This isn't the equation for any two masses. This is for two spherical or point masses. The equation between a rod and a sphere would be different. You can get creative with different geometries.
As others have said, you can plot the force as a function of separation distance.
http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
Just look up the values for the masses of the Earth and Moon and the separation of their centers of mass.