It all works nicely until one point, then everything looks like on the image (both in edit and object mode). Even when I save and reopen the file. :S
The issue is that you set the clip range to a very large value:
By setting the end clip to such a large value you are loosing precision. Blender can no longer resolve geometry that is close to each other. The issue is called Z-fighting, the same as when you have geometry that is occupying the same space in the scene (This happens in all 3D software not only in blender). As a rule your start and end clip settings should be so that they encompass your scene, but not a lot more. Why would you need to set the end distance to trillions of Km when you are working with objects are a few centimeters long?