Distorting a surface with points or curves

I am trying to make a surface that gets distorted based on the location of either points or lines. So what I have now is a surface that gets created based on the location of a single point in grid. I randomized the list of distances from the single point to the grid points to distort the surface. 

1) Is there a way to give what I have a variation so that it creates various iterations? Now it only changes proportionally. Also is there a way to apply multiple points on a single surface so that if I add more points, it creates a different shape based on the location of the new points?

2) I was initially trying to make a surface that gets distorted around where the point is located. Something like the image file, but in 3d. (It would be nice to have something like that in 2d too) Anyways I couldn't figure this out. This would also be nice if multiple points work as an retractor.