In legal proceedings there seems to be a nebulous entity often referred to as "the Court". What does this device/construct denote and what is its purpose? Is it basicallyy like an avatar that the various judges of the court can hop into the controller cockpit of, to accommodate for the fact that it is not always the same judge dealing with any given case each day?
It is the court of law processing the particular legal proceeding in which the reference is made.
Typically it is unequivocal which particular court (both the level and the geo location) is talked about because the title sheets of the documents filed in proceedings say it explicitly.
Depending on the context, "the court" may refer to either the official organisation (as a node in the justice system), or — quite often — directly to the judge(s) presiding in the proceeding.
The latter is why, for example, formal documents filed in proceedings typically start with "MAY IT PLEASE THE COURT:" or "MAY IT PLEASE YOUR HONOUR:". The latter means exactly the same as the former where there is only one judge presiding.
When there is a need to refer to some other court than the one handling the proceeding, it will be referred to more specifically. For example, when going through a first appeal, the court which made the decision being appealed against is referred to "the trial court" or, say, "the District Court at place XXX". When going through a second appeal, the reference will be "the 1st appeal court" and so on.