The nature of the Trinity and of Mary

I was watching a Catholic show on early heresies last night, led by a theologian whose name I can't recall. The host dealt pretty harshly with heresies specific to the nature of Christ, and it drummed up some thoughts about when I was a believer - I didn't really understand what the Incarnation actually means.

Unfortunately, the only noted heresy I recall by name is "Arianism," due to its similarity (only by the sound of the word) to the racial policies picked up by the Nazis. The concept, banned as heresy by the First Council of Nicea, taught that God created Christ at the moment of His birth, if I recall correctly.

The Catholic theologian went on to explain that God is one essence and three persons - all of equivalent natures, not just similar ones. How can this be so? It is not entirely clear to me, especially in light of John 14:28, how they can be considered equivalent, but maybe I'm missing something here.

I simply can't understand what the doctrine of the Trinity means. When I was a believer, I thought of it as a corporation, the "God corporation," in which three separate Beings filled the necessary roles to comprise a coherent notion of God. Are these three Beings separate, or not? Are they equal, or not, or would the term "equal" be used in different senses here?

One last point, especially considering any believing audience of mine is a Protestant one: if Christ is equivalent to God, this makes Mary the literal mother of God. It seems she's due the veneration (but not the worship) that Catholics give - I actually agreed with the show's host on this point. When did such veneration fall away in Protestant thinking, and why?

Just some thoughts - looking forward to your responses!

-Darrin