RR - what I'm saying is, I don't think much of that is distinctively medieval (bar Anselm). I don't think anyone who argues this stuff today has advanced very far.
But I'm pretty sentimental about medieval religion because I work on it and it's hard to read people being very earnest and thinking carefully about things, without wanting to respect them even while you disagree with them.
animula: basically, in the early centuries of the Church, people were trying to think what 'God' might mean. As they understood it, Jesus had been revealed as the Son of God. And they'd agreed that there was a Spirit or Holy Ghost, which was another way in which divinity could make itself known to mankind. But what people couldn't agree on was, how were these elements of God related to one another? You see, many people in those days were familiar with the idea of many divine beings (ie., polytheistic religion). It must have been difficult for early Christians to explain that they believe in only one God - but they thought this God took three forms.
The Arian heresy, as I understand it, describes one of the attempts to define the relationship between the three persons of God. The established Church eventually decided that Arianism got the balance wrong, and didn't accurately represent the relationships between the three persons of God.
This issue about how you describe the relationship between the three persons of God is historically important, because it was one of the reasons for the split between the Eastern and Western Churches (the 'Great Schism'), during which the Eastern Church rejected the authority of the Pope.
I'm less sure about Newman (sorry if that is the bit you wanted to know about most) - I find it really hard to get into the mindset of Anglo-Catholicism. But I think his argument with the Anglican Church had as much to do with his concern about the authority that Church had, as with anything else. Basically, there will always be a question mark over the Anglican Church's claim to be an original church with its own authority, but I think that seemed a radical idea in the Victorian era. As I understand it, he thought the Anglican Church had been too tangled up with man-made structures, whereas the Catholic Church might be closer to what Christ revealed.
But I get immensely confused by Victorian religion in general, so I expect there's a lot more to it!