The man walking from the bridge toward the cathedral is wearing a baseball cap, but Castle is not wearing a cap.
Detective Beckett comments that Agent Harris' estimation of the search area (30 sq. kilometers) is the entire city of Paris. In reality, Paris has an area of 41 sq. miles (104 sq. kilometers).
When Jason Hunt shoots the men in the woods, as well as Castle's phone, the gunshots are loud and crisp, but the rifle he is carrying has a huge silencer on it which would have muffled the sound at the least, and completely silenced it if working properly.