True! I did some research and the three-wins-and-keep-the-trophy was at the behest of Mr Rimet for the original trophy. The current trophy doesn’t follow this rule, in fact the FIFA rules explicitly state this apparently.
West Germany was just a tag used to differentiate the two Germanys; since the war the country has always been the Federal Republic of Germany, and the team which won the WC the 4 times was representing the FRG. They have never been two separate countrie.