Quite a bit of genocide occurred in the territories, but most of that calmed down before they reached statehood.
The genocide and marginalization against the native population continued well after the union, as far as I know.
Did the creation of the EU stop some war in Western Europe?
That is commonly believed, yes - it has brought together countries that had been in a bitter war less than 50 years earlier, and may well have preserved the peace.
Pure speculation. Just like the states haven't had a war because they're the United States. Imagine if the states would have become independent countries. I can imagine the endless warfare. Nobel in Peace as clearly deserved as the European States.
Sure, it's all speculation, not claiming otherwise. But this can be said about any historical event that was "prevented" - by definition, it didn't happen, so we can only assume that it could have happened if something didn't stop it.