If you've ever wondered why many US cities have French names (Baton Rouge, Des Moines, Boise, Terre Haute, St. Louis, Louisville, Dubuque, Detroit, Marquette, New (Nouvelle) Orleans), it's because those were all under French control when they were named.

The colonists couldn't expand westward without hitting French territory, so yes they wanted war against the French.

The British settlers along the coast were upset that French troops would now be close to the western borders of their colonies. They felt the French would encourage their tribal allies among the North American natives to attack them. Also, the British settlers wanted access to the fertile land of the Ohio River Valley for the new settlers that were flooding into the British colonies seeking farm land

https://en.wikipedia.org/wiki/Seven_Years%27_War

Do you know what actually started the 7 years war? It was when George Washington (a Lt. Colonel in the British army) ambushed a French force who were building a fort (Fort Duquesne) to defend their territory near the Ohio river. The French then attacked Washington's army and forced it to surrender. The first battlefront in the 7 years war was in North America, and it was a territorial dispute over the Ohio river valley.