Funny thing: The Ottomans asked me to declare war on Montezuma and I accepted. After that, four city-states decared war on Montezuma LOL. Probably Ottoman Empire's allies.
Yes, if you declare war on someone else, your city states will follow. So if they had city states and they declare war, their city states will do so as well, and if the one who is attacked has city states, they will also declare war on the ones attacking them. As if they were a part of your empire.