By the time of William the Conqueror, which I think is the sixth generation of Rollo's line in Normandy, they were just French (with a cultural memory of North Sea origins). The tapestry of Bayeux, which was made in England, calls them that.

Its a mess. Vikings (mostly danes) did "move" there by conquering and being given lands as bribes. William conquered England but was still a vassal to the king of France due to still being the Duke of Normandy. So for example when France got a new king the king of England would need to go and swear loyalty and such, which would become a problem later.

Through marriages and such the Duke of Normandy took over large parts of France and it became the Angevin Empire, but still just a puny vassal to the King of France.

The 100 year war was fought over this essentially and England would end up losing all French land and thus the problem was solved forever.

yes, but when they moved to England, the Normand duke made himself king of England, so he (and his heirs) had the crown of England and the ducky of Normandy.