The one or two times we saw the Federation Council it seemed like just that, a council. A group of different powers that tries to work together from time to time, kind of like how I imagine the UN is supposed to work. Of course, not having a political degree I could be completely wrong on that.
Well, I
do have a degree in political science, and I can tell you the following things:
1. A "council" is just a deliberative body of any sort, and is not inherently comprised of representatives from multiple sovereign states the way you seem to be implying. A council can be a governing body for a private organization like a church, or it can be the legislature of a city, or it can be the legislature of a sovereign state, or it can be a governing body of an intergovernmental organization.
2. The Federation government possesses all of the traits of true state, not a mere intergovernmental organization like the U.N. It had an elected President (ST4, ST6, "Homefront"/"Paradise Lost"), it made foreign policy for its members (ST6), it had a legislature that could make binding law in its territory ("Force of Nature"), it had territory ("The Best of Both Worlds"), it had a military ("Court-Martial"), it had law enforcement (Federation Security, seen in ST3), and it could declare war ("Errand of Mercy") and martial law ("Homefront"). None of these things can the U.N. do. The United Nations is not a sovereign state; the Federation is.
* * *
As for the existence of Earth countries within the Federation...
First off, bear in mind that we've been using the terms "country," "nation," and "state" interchangeably here, but they're not the same thing. A "country" is a distinct area of land. A "nation" is a group of people who share a common identity and, typically, a common language, history, and descent (real or imagined). A "state" is a sovereign political entity possessing its own distinct territory over which it has exclusive legal jurisdiction to make law and use force.
Thus, Earth's countries will always exist so long as Earth does, just because the land is still there.
But, to get a bit more serious, there's no reason at all that the sovereign states that exist on Earth today could not continue to exist as non-sovereign political subdivisions of United Earth within the larger Federation. The levels of government for, say, someone living in Toronto could look like this:
UNITED FEDERATION OF PLANETS
President, Council
UNITED EARTH
President, Prime Minister, Parliament
CANADA
Monarch, Governor-General, Prime Minister, Parliament
PROVINCE OF ONTARIO
Monarch in Right of Ontario, Lieutenant Governor, Premier, Legislative Assembly
CITY OF TORONTO
Mayor, City Council
For someone living in, say, the City of ShiKahr on Vulcan, it might look like this:
UNITED FEDERATION OF PLANETS
President, Council
CONFEDERACY OF VULCAN
Administrator, Council
PROVINCE OF PELASHT
Minister-President, Assembly
CITY OF SHIKAHR
Mayor, Council
Meanwhile, for someone living in, say, the capital of Andor, Laibok, it might just be:
UNITED FEDERATION OF PLANETS
President, Council
ANDORIAN EMPIRE
Chancellor, Parliament Andoria
CITY OF LAIBOK
Mayor, Council