Is a FAA certified plane allowed to fly everywhere in the world?
It's valid in countries that choose to accept it as de-facto valid but sovereign nations are free to require local certification if they like. The reality is that the FAA has a huge amount of resources at its disposal and for smaller nations is advantageous to simply say we agree that those people know what they are doing and we can't afford to do it any better. 

However Boeing does hold an EASA Type Certificate for the 737 as well as the FAA issued one but if you dig into it you will find a lot of "see FAA TCDS" notes. With that said the 737 is certified by European agencies currently. A good chunk of regulations even stem from the FAA work, the best example of this is in the EASA TCDS where EASA changed some regulations to the FAA spec. 


  However, to harmonize with the FAA, the same requirement was moved to JAR 25.810(a)(1)(ii) at Change 14 which is now in
  CS25.810(a)(1)(ii).