"What determines the number of points on a surface when you click show points?"

It's the number of control points of that you have yourself created when you have drawn your curves ;)
So your number of clicks if you have used a mouse ;)
Then multiply when you make a command betwen these curves (Sweep for example)
And surely yes the minimum for the surface will be the more approached according the curves

For the sphere I believe the number of points is the less reasonable number for a primitive for future rich deformations ;)
Because a sphere's surface can have only 28 control points !