Will I need to show my insurance documents when I cross the border into the EU? Will I also need to show the documents when I move from one EU member state to another? 

You may be required to show documents at the border when entering the EU, but this will be a decision for the border authorities to take. You may also be subject to police checks while driving abroad and you will also need to be able to present the document at the scene if you are involved in an accident.