"Don't add space between paragraphs of the same style"

It sounds to me as if you want to remove the Spacing After from the Normal style entirely. Doing this would be more efficient than enabling "Don't add space..." If either of these changes is not being saved, however, then it is likely that an add-in is preventing the Normal template from being saved. (One common culprit is the PDF Complete add-in, which comes installed on many HP systems.)

The permanent solution is to determine which add-in is causing the problem and eliminate it, but a temporary workaround is to open the Normal template for editing, make the change directly there, and then save the template. Here's an easy shortcut to do that:

In Word, press Alt+F11; the Visual Basic Editor opens. Press Ctrl+G to activate the Immediate window. In that window, type (or paste)

     NormalTemplate.OpenAsDocument

and press Enter. Normal opens, but note that it may not be on top of all windows. You can close the Visual Basic Editor. Make the desired changes to the Normal file, and then save and close it. If the changes you have made have not been recognized as changes (the Undo button is not activated), press Spacebar, then Backspace to "dirty" the file, thus ensuring that the template will actually be saved.