In what sense is God one; do the three persons share one body?

No. God does not have a body as we do. The Father, Son, and Holy Spirit are three distinct persons made of one and the same divine substance. God is one in substance, not in body. He has what we call divine substance. Each person of the Trinity is His own person, with His own personality, yet each is made of the same divine substance. That may look like a contradiction. It is not. I am made of human substance, and so is my wife, but we are not the same person. The analogy only goes so far, because two humans are two beings, while the three divine persons are one God. How God's one substance holds three distinct persons, we do not know, and we probably will not know until we get to heaven and God explains it to us. If you want the detail, read the Council of Toledo. It was a local council, but the Church approved it, and it says more about the relations within the Trinity than any other council. You will find it in Denzinger, translated by Roy Deferrari as The Sources of Catholic Dogma. Athanasius is the best place to start among the Fathers.