After William conquered England and became king was he still under the rule of the French? Or at least the land he controlled in France? I've heard that the English King would attend the King of France's coronation and would swear fealty to him, but I am not so sure if it is true. Another question is why did the French King not have at the very least nominal control of England, as their king should have been a subject to the French Monochary.
It got more complicated later on in the Middle Ages. Where the English Kings were Dukes of Aquitane. But what's important to remember is the French King had authority over the duchy of Aquitaine, so the English King had a duty to swear fealty AS the duke of Aquitaine, but not the King of England. They were not one in the same title, so the French King had 0 power over say Edward III as King of England, but technically had power over him as duke of Aquitaine, but only on matters related to Aquitaine. Technically the King of England was liable to provide soldiers for the King of France, but they never called upon the English for obvious reasons.
Yes, William was a subject of the French throne, a fact that caused him and his descendants no end of headaches, which led to multiple wars, and was a major contributor to the famous franco-anglo rivalry.
Mary Sterling's Clash of Crowns is an excellent book on the subject though the main focus is Richard the lionheart, it definitely goes into the context that created the situation in the first place.