Is the CP that appears before the auxiliary moved up from a lower level?
No, I don't think so. Where would it be moving from? There is no lower position.
Raising as a technical term only refers to NPs as far as I know. For example:
John seems to be happy <= seems John to be happy.
Movement (in general) can occur for CPs as well, but only if there is some lower place from which they originated.
What is the status of the C.P in the first sentence? Subject?
Yes, it's the subject of "is important".