This question gets asked a lot, and the answer depends on who you ask and in what context. Since the High Church is a multi-faceted organization, incorporating the beliefs of many different earth religions and cultures, different branches will have different answers.
For example, the Karmists believe that the Emperox is a piece of the divine soul, incarnated into human flesh. The Purists believe that the Emperox is the prime ideal, the perfect human, and in that way becomes "divine". Some others may yet hold to the idea that the Emperox is not technically a god, but is sent on a holy mission to rule humanity.
What all branches hold, in the end, is that the Emperox is "divine" (whatever they mean that to be), that they have the right to rule humanity, and that this right extends from the divine (either the Divine Right of Kings or via the Will of Heaven, read their respective wikipedia articles for more information).
The Church's primary aim in relations with the other faction is to further stabilize the Empire. The Church sees its goal as serving as the anchor. While the other factions decide who will rule, what to do with non-imperial forces etc, the Church focuses on making sure the essential services the Church provides are provided. In so doing, they seek to make sure the Empire doesn't fall into anarchy or civil war.
No. Really, no. We aim to stabilize the empire, so unless your faction threatens to undermine that stability it is highly unlikely the Church is trying to kill off your faction. Even during the synth war, the Church didn't land troops on Cygnus-held planets. Nothing said IC should be construed to mean that we are "going after" a faction. We express our political ambitions purely through the faction turn and official diplomatic channels.
That is actually a very difficult question, one that we have not completely sorted out, so this answer may be edited at a later date to better reflect current thinking and rulings. However, here is what we have so far:
No, you give up your nobility to enter the clergy. However, your status will be evident in your behaviour, family, contacts, upbringing, personal wealth and other such identifiers that mean the serf/noble distinction still exists informally within the ranks of the clergy. In rare cases, a serf may find themselves rising to be on par with nobles in power and influence. They'll still never really be viewed as a respected noble peer.
No, that's the wrong sense of the word. When we named the Order, we were under the impression that we would have sort of ecclesiastical law enforcement unit, not a gang of guys (and girls, girls can torture too) running around the sector burning people at the stake.
The Inquisition is more like the ecclesiastical FBI. They maintain investigations on threats to the Church, the Emperor, or the spiritual wellbeing of the Sector.
No! The Emperox is chosen by God
Yes, and it is understandable that many will have Messiah and God meaning the same thing due to the culture of Trinitarian Christian beliefs, and people thinking 'Jesus' when they think 'Messiah'. Messiah means 'saviour' and there are many people known as a Messiah throughout religions today. A more apt analogy for the Emperox would be Moses or King David.
Yes! One of the theological problems that the setting has handed to us is a Messiah that has done things the Church does not support. Technically we should be following everything they say. We had the same problem with the Blood Eagle, and other "Bad" Emperox, where we have to publically support their actions regardless or have our belief system suspended.
The assassin was secretly approved of by the Church, but we are in a difficult position. If we approve of someone killing God's chosen, we are casting into doubt the Church's spiritual authority. Clearly we would then be incapable as an empire of choosing the correct Messiah. However, if we go on to punish Serpens, they know that we secretly approve! They have the pre-cog council who have saved the empire many times. They are an ally of the Church. We are stuck between two hard places and it will not be possible to find a "good" solution.
We are Pro-Human, with Anti-Alien being secondary. As part of our teachings as Humans being the only creature with a soul we therefore must be the best creature. Aliens that are no threat are allowed to exist. They range from beloved household pet, to livestock, to vermin.
Synthetic animals do not challenge the idea of Humans being uniquely ensouled. The problem with Synthetic Humans is the intelligence, behaviour and skill shown making people question if Humans are that unique after all. They call into question the power of God, for if Humans can make a ensouled creature.... (Synthetics are of course, identical to humans and the idea of a soul is religious guff to make us all feel unique and special.)