We seem to meet an awful lot of mostly non-recurring admirals.
What's the command structure at the top of Starfleet?
Does Starfleet report to a Commander-in-Chief? Is the Federation Council involved? Is the President involved?
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The Memory Alpha page on Starfleet has a section specifically on its organisation. It describes it as follows:
Direct commanding authority over Starfleet was a power of the Federation President. (DS9: "Paradise Lost", Star Trek VI: The Undiscovered Country) In addition, the Federation Council, the Federation's legislature, often directly influenced the service's operations and even issued orders on occasion. (TNG: "Force of Nature", "The Defector") The day-to-day operations of the service were managed by Starfleet Command, under a commander in chief. (Star Trek VI: The Undiscovered Country)
As for the command structure itself, we have several important figures:
The commander in chief oversaw the operations of the entire service, and reported directly to the Federation president. (Star Trek VI: The Undiscovered Country)
This was the service's operational authority. Senior positions included the Commander, Starfleet. the chief of staff and the chief of Starfleet Operations. (Star Trek III: The Search for Spock; Star Trek IV: The Voyage Home; Star Trek V: The Final Frontier)
These oversaw operations of the service within sectors of Federation space. There were at least 23 sector commands. Sector commanders reported to Starfleet Command, and commanding officers of service vessels operating within these sectors reported to them. (TNG: "Power Play", "The Measure Of A Man")
In addition, there are a whole host of agencies within Starfleet. So, in answer to your questions:
The President is referred to as having been elected, suggesting a civilian position. We can interpret these facts in either of a couple of ways:
The difference between the two scenarios appears academic: senior Starfleet personnel lead Starfleet, and answer to the civilian government, sometimes embodied by The President.
As for The Federation Council, it is generally depicted as a legislative body. Its edicts are imposed on Starfleet, but we have yet to see Councilors issuing orders to Starfleet personnel. The workings of Federation government have been left vague in canon, but an American-style separation of powers seems to be in effect.