[Natural Law]

"So, does the Catholic church teach that there are three gods together who form the godhead since one is not the other, or does it teach that there is one God expressed in three different ways, since there is one God and they are each other, or both or neither?"

The Church teaches that all three members of the Trinity are God, not Gods. The Trinity is a mystery of faith in the strict sense, one of the "mysteries that are hidden in God, which can never be known unless they are revealed by God". To be sure, God has left traces of his Trinitarian being in his work of creation and in his Revelation throughout the Old Testament. But his inmost Being as Holy Trinity is a mystery that is inaccessible to reason alone or even to Israel's faith before the Incarnation of God's Son and the sending of the Holy Spirit.

[metmom]

Good answer. That answers all the questions at once.