How does the church relate to the doctrine of God/the Trinity?
Component question: which element of the Trinity does the church come from?
This is a dogmatic structure question, and probably rhetorical, but it is also fundamental to how you understand the church.
It is a common Christian perspective to link the church to the Son, as Jesus is seen as the beginning of a new phenomenon, and/or also to the Spirit, as the typical magisterial position holds that the Spirit guarantees the position of the church. Of course, with less cynicism, Luke-Acts is quite clear that God, through the Spirit, works directly in the world to secure the divine purposes (and the church).
From a contemporary perspective, it is easier to see the ongoing church as the work of the Spirit. It is easier to see Jesus as the origin point. The Gospel writings show Jesus in his earthly career, gathering messengers and sending them out; Paul writes very clearly that his mission is oriented solely on Christ. There is an inherent discontinuity in this thought that isn't in the writings, I think. It comes out of the break with the Jewish sects, and it's problematic to how we are, in that image, people of God. I've started reading Wingren, and he reinforces this. The Bible is predicated upon the unity of the two testaments, and their reference to one God. What does not seem to come out is the idea that there is a unity to the people with whom this one God covenants.
Obstacles: I'm not talking about Jews becoming Christian, and I'm not talking about Christians becoming Jewish. I'm emphatically not talking about Christianity superseding Judaism. Theologically speaking, those are dead ends, and beside the point anyways. What I do mean is that, for consistency, the doctrine of the church must relate to God in all three persons. We're missing the Father.
We recurse: what is the church? If the church is the people of God, and God has had a people at least since Abraham, then at least there is a logical unity in which the tribes of Israel and the apostolic communities belong. We are people with whom God covenants. Now, it helps to have the intertestamental literature, but even the Gospels will do for this: it is easy to see, historically, that the followers of Jesus come out of the Jewish tradition. Jesus was a Jew. The "scriptures" of the early church are the Torah, the Neviim, and (possibly) the Kethuvim. Obviously, 70AD and the LXX usage conflicts get in the way, but they do so because, sociologically, the communities had already begun to divide. God doesn't stop covenanting -- God is faithful. God extends his promise beyond the boundaries of the fragmenting communities to the nations. That's how we got in. It must therefore be said that the existence of the church within "people with whom God covenants" is due to the action of the Father as much as the Son and the Spirit. Our covenant is with the Father.