I’ve been playing Crusader Kings 3 a lot (I know it’s not completely accurate) but it got me wondering if the duchy of Normandy was considered part of France after the conquest of England and what the politics were like between William and France during that period.
Following the decline of the Carolingian dynasty in West Francia, the kingdom experienced a collapse of central authority. The ensuing decentralisation led to the rise of hereditary duchies and counties that nominally answered to the king but were independent in practice. As a result, there was no such thing as "being part of France", what distinguished the dukes of Normandy and the count of Anjou from their counterparts in the Holy Roman Empire was the object of their fealty. Said fealty didn't equal loyalty, however; it didn't prevent them from taking up arms against the king's armies several times.
After becoming kings of England, the house of Normandy and their successors, the house of Anjou, continued to rule Normandy through fealty to the king of France. But in practice, they were as rebellious as they were before obtaining the crown of England, and in the Anglo-French wars of the late 1100s and early 1200s, would fight against their liege alongside the counts of Flanders and Boulogne, also nominal vassals of the French crown.
For this reason alone, it is debatable that those duchies and counties were ever "part of France"—they were full states with their own internal and foreign policies, and sometimes even their own currency. Yes, Edward I did formally pay homage to Philip IV for his possessions in Gascony, but such formalities have little incidence on the actual political situation. In the case of Normandy, it wasn't until the resurgence of royal authority in the 13th century and the ensuing conquest led by Philip II that it can be considered part of France.