It is a bit unclear how you got the data representation you are using. You stated that the file started as an .stl file. Stl files represent 3D geometry as a tessellated (faceted) model. That is, all surface are modeled as triangular facets that define a completely enclosed volume. Somewhere along your workflow this model was changed to circles and defined as blocks as shown in your image Sphere.png. IF, as @Alfred.NESWADBA notes, the block insertion point is at the center of the sphere you do not need to add point objects to the AutoCAD file. Just extract the block locations. If you find you need to add point objects then:
1. Create a layer for the points (e.g., Sphere Centers) and make it active.
2. Set the point style to something you can see with the ptype command (I don't know why the default point style is something you can hardly see).
3. With osnap set to center, add points at the center of the "great circles" that are part of the sphere (from the image you supplied it looks like they are in your model).
4. With all layers frozen except the layer with the points give the dataextraction command. Note, you can easily freeze all layers except the current active layer with:
-layer [Enter]
f [Enter]
* [Enter] [Enter]
5. Follow the steps presented by the dataextraction dialog. You probably only want the Geometry Category filter. The dataextractrion process is fairly user friendly.
AutoCAD has many idiosyncrasies that take time to learn. For example, a hyphen in front of a command invokes a command without dialog boxes. This allows you to use the keyboard exclusively without need of the mouse. That is why my instruction above have you give the -layer command and not just layer. Regarding layers, there is a difference between a layer that is frozen and one that is off. It's generally better to freeze layers than just to turn them off as off layers may still be affected by AutoCAD operations.
Good luck.
lee.minardi