We know for a fact that the Federation is a democratic body from Kirk's line establishing such in "Errand of Mercy." We know the Federation President is elected from "Paradise Lost." There is also a Federation Supreme Court, established in "Dr. Bashir, I Presume?," and a presidential Cabinet, established in "Extreme Measures." This much is known canonically.
However, voting was part of the process where Ambassadors (and possible other officials and reps) decided whether to accept Coridan's petition for UFP membership in "Journey to Babel". The vote was to be cast in a special emergency conference, which probably isn't the same thing as the UFP Council, and it appeared every member planet got one vote. This might be considered representative democracy at work, if only we had some evidence that the Ambassadors casting the planetary votes represented their people via a democratic election process.
To be fair, "Journey to Babel" estbalishes that the admissions process for Coridan was an extraordinary situation for the Federation, with the Member worlds at each other's throats, to the point where violence seemed likely to break out. I'd say that that represents a clear constitutional crisis, and shouldn't be taken to represent what normal Federation processes look like.
Well we know very little about how the Federation Council works. Only what can be inferred from the vague referrences in episodes.
For whatever it's worth, the novels Articles of the Federation
and A Time for War, A Time for Peace
have done a lot to establish how the Federation government works.
Each Federation Member state has a single Federation Councillor to represent them on the Council. That Federation Councillor is appointed in whatever manner the Member state choses; the Federation Councillor from Betazed is popularly elected, for instance, while the Federation Councillor from Andor is appointed as part of the Presider's cabinet on the basis of which party wins a majority of seats in the Parliament Andoria, and the Federation Councillor from Bajor is appointed by the First Minister with the ratification of the Chamber of Ministers.
Federation Member states still retain limited capacity to conduct direct bilateral relations with one-another, and with foreign states, under Federation government supervision. As a result, each Member state retains an embassy on Earth, and a mostly-ceremonial Ambassador to the Federation as well. The Ambassadors seem to be almost completely powerless; presumably, they're appointed by the Member state government and answerable only to it, and function merely to facilitate relations between the Federation government and Member government, in contrast to a Councillor, who is an employee of the Federation
government even as they represent a Member. Think of it as the difference between a U.S. Senator and the governors' delegations that some states send to D.C. to represent their economic interests in Washington.
The Federation President is directly elected by universal adult suffrage. Petitions for candidacy are submitted to the full Federation Council, which then votes to confirm that a given potential candidate fulfills the legal requirements for the Presidency, granting them official candidate status. The process of counting all of the votes for the Federation President takes around one week, and is conducted by two independent auditing firms as well as the Federation government itself, to ensure accuracy.
In the event of a Federation President's sudden vacancy from office, the Council appoints one of their own as Federation President Pro Tempore, who governs for one standard month while a new election is called.
The Federation Council is divided up into numerous committees, referred to as sub-councils. Each sub-council has authority over bills dealing with a given area. The Federation President appoints each member of the sub-councils, with the ratification of the full Council. The Federation President is expected to preside over all meetings of the full Council, and has the option to preside over all sessions of the sub-councils. The President works very closely with the sub-councils when formulating policy, particularly the Federation Security Council (which has jurisdiction over areas related to national security). In addition, departments of the Federation government are funded directly by their respective sub-councils, to whom they answer in addition to the President. Bills that are passed by a sub-council are then passed by the full Council, and then are either signed or vetoed by the President. If vetoed, the bill goes back to the full Council for a potential override vote.
The Federation Judiciary Council is depicted as the highest judicial body in the Federation. This may be interpreted as contradicting the line establishing the Supreme Court in "Dr. Bashir, I Presume?". However, one might interpret the "Dr. Bashir" line as referring to the Judiciary Council; certainly the Law Lords in Britain, until recently, were something similar.
So it's a bit of a hybrid of the Presidential and Parliamentary systems. There's much less separation of powers than in the U.S. system.
We don't know for definate even how the United Earth Government works, from what can be inferred from ENT is that it appears to be a Parliamentary based, but is that with an elected head of State (President) with a Prime Minister being the Head of Government.
That's exactly what the novels have established. The United Earth Prime Minister is established to still exist as of 2375, and in the ENT novels, we learn that United Earth is governed by a Parliament, with a President as a mostly-ceremonial head of government and real power lying with the Prime Minister.
I would definitely bet on this. The acts of voting and campaigning and all the oversight involved just seem way too chaotic for the Utopia.
You have a strange idea of "utopia," if you can't even vote for your own leaders. Sounds very dystopian to me.
Of course, we should let go of the idea of the Federation as utopia. The Federation is not a utopia, it's just more progressive than what exists today.
Not to mention it seems logistically impractical even with their technology.
You don't think that a culture that can travel faster than the speed of light has figured out how to count
It also seems to me that such things like demanding to have a vote in whatever would look really petty to them in the future.
Oh, yes, all those petty, petty people, like Alice Paul and Martin Luther King. So petty, this desire to have a voice in the government that has power over you.
I'm sorry, but that idea is just naked autocracy.