you can subdivide the caps of your extrude generator. down the bottom of the caps tab under caps type . hit the tiny little grey arrow next to the caps type text for more options you can for example set it to regular grid then alter the size until its as clean a mesh as you can get.
does the object need to stay as a generator or can you convert it to an editable state?
As to cutting the mesh if you use the plane cut tool you can quickly create a grid.
Long days and pleasant nights.