A similar thing happens on Yom Kippur. Kids are there for the whole service. No one has planned ahead to make some kind of arrangements for them such as to take them out for a walk or have set up a room where they can take a nap. Inevitably, they make a nuisance of themselves.
As far as kids thinking that you are a goy, I have heard kids ask this because the person was taller than their Tatie, or because they had red hair or black hair. Their measuring stick for all things Jewish is their own home, so if their Tatie is 5’5” and blue-eyed, then those who aren’t don’t fit the mold. The parents are at fault if they observe this, and make no effort to explain that frum yidden don’t all sound or look alike.