Is just war an actual doctrine of the Catholic Church, and how do we know something is Church doctrine?

Yes, it is actual doctrine, because it has been used many times in the history of the Church. The world used to go to the Church for the moral basis of things like taxation, war, slavery, and marriage; kings would ask popes, as we see in the writings of the fathers, where Pope Leo the Great answers bishops from all over the world. War was at the top of the list. A war is unjust when a nation aggressively attacks another to seize its resources, gold, water, and so on, instead of waiting on trade; the nation being attacked may justly defend itself, its people, and its resources. The popes made these decisions based on the saints and doctors, Thomas Aquinas and Augustine being the big ones on just war. A papal declaration that a war is just also gives other nations reason to come to the defender's aid. So yes, we could not live without just-war doctrine; it is a necessity.