You’re probably asking too many things at once. I mean, you just asked for a precis of 500 odd years of church history, +/- 200 on each side, and diverse theological positions and controversies. It’s not genuinely worth my time to try and write a mini-description of each of these positions, when in the space I could allot, you could go and read a half-decent Wikipedia article on each of them. Most of them make much more sense if you have some kind of grasp of the history and theology of the period to make sense of why the debates arose and what was at stake.

What might help is to realise that one of the features of Christianity in LA was that it emerged as a religious system that hard a very large stake in the ground over the issue of truth. It mattered what you believed, and if you got it massively wrong then you were heretical, outside the Church, and facing eternal damnation. Similarly, if you got it wrong then you were basically believing and teaching wrong things about the God of the Universe and so that was something pretty big to be wrong about.

For example, in the Arian debates, the argument largely centres around who, and what, Jesus is. What does it mean for Jesus to be ‘God’? For Arius and those that followed him with similar ways of thinking, he is the first and foremost created being, who by his exemplary existence and example paves the way for salvation for the rest of us. That was a system of understanding both the metaphysics of Jesus (what he is) and soteriology (how salvation works). But, on the other side, the pro-Nicenes (that’s what we tend to call the victorious side in contemporary works) said that this system actually doesn’t work, it doesn’t match with what the Bible says about Jesus, and if this was “how it was” (i.e., if the Arian position was true), then salvation became unworkable, because a creature like us could never save. (I have omitted a lot here).

The pope cares because the pope in Rome is a player in a theological and church-political battlefield. In the first few centuries of this period, the pope isn’t the major power that he is later on. But regardless, the pope cares because just as in society, so in church there are politics. Politics and theology were never that far from each other, because politics is just human life. Anyway, this gives the pope two reasons to play the game, to engage in theological controversy – to get things ‘right’, and to outmanoeuvre one’s opponents.

A similar dynamic can be seen for Emperors, although the weight is often more on the political side. Firstly, in Christendom you need to have the church on-side for good government. But then you discover that within the church, and outside the church too, are various groups and factions that want your support for their positions. So as a political figure who is also seen by many as the leader of the Christian world (unless you had the fortune/misfortune to live outside the Empire, say in the middle east), it’s almost inescapable that you will be drawn into such issues, and need to decide who you favour and whom you don’t favour.

That’s a start. Do you want things more specific? I don’t really know what your question is trying to drive at.