The Angevin Empire was not a country in the way that we view countries today. The modern notion of a country didn't begin until after the medieval era had ended.

It's easier to understand the organisation of medieval kingdoms if you view them as companies instead of countries, and view the feudal system as a monetary system instead of as a military hierarchy. Henry II held various separate titles such as Duke of Normandy, Count of Anjou, King of England. You can think of these as separate jobs. Which happened to be held by the same person, and he was paid for performing these jobs by being given land.

During the medieval era there was only a limited amount of money in circulation, and kings did not have the money to pay their soldiers. So Kings did not pay their Barons in money, but instead paid them with land. The Barons then paid their soldiers with land.

The system of paying people in land wasn't restricted to the military, civilian institutions such as monasteries were also paid in land. Originally the land wasn't given permanently but would only be given to someone so long as they performed the job they were hired to perform. However since people rarely changed jobs, and jobs tended to be hereditary by this point in the middle ages, the vassals essentially held the land as if it was their own land instead of the land of the king.

Henry II's ancestors had been given land in France in exchange for serving the King of France, and he held that land in France so long as he promised to continue serving the King of France. He was King of England in his own right irrelevant to any service he made to the King of France. The Kingdom of England was never a vassal to the Kingdom of France, the King of England just happened to be someone who also held titles that were vassal to the Kingdom of France.

Henry II ruled over England separately from France and the laws of England and the laws of France were separate from one another, and in France there could be a significant amount of variation in the law from one area to another. A merchant in England importing goods from Anjou would need to pay customs duties on what he imported. Economically the fact that they were ruled by the same people didn't matter.

As for military matters, legally speaking since these titles were separate from one another them being held by the same person should not have made a difference. However in practice during medieval times royalty and aristocracy rarely followed the law if it couldn't be backed with force, so the French King was worried that Henry II would expand his power in France, and possibly overthrow him, so he wanted to limit Henrys' power, prevent him from taking more land and take his land away from him.

So Henry needed to stay in France so he could prevent rebellions by his own vassals in France, and so he could respond if the French King tried to take his land away from him.

As for what happened in Toulouse, Henry II was married to Eleanor the Duchess of Aquitaine. Eleanor had a claim to Toulouse because she was a descendant of William IV Count of Toulouse and according to the law should be the Countess of Toulouse, however her family had not controlled Toulouse since 1094.

Henry II wanted to retake Toulouse and expand his power Louis VII wanted to prevent him from doing this so he defended Toulouse against Henry. Henry didn't want to go to war with Louis, so he didn't attack Toulouse.