Re: Did we ever see anyone vote in the Federation?
A week to count all the votes, in some countries the result of an election is known within 12hours of polls closing. And that's wiith paper votes.
The largest of those democratic countries (India, population 1.2 billion) has a population that's likely something in the area of 0.12935051111% the size of the Federation's population.
[This is assuming 155 Federation Member states and an average Federation Member population of 6 billion; an average Federation Member colony population of 12 million and an average of 3 Member colonies per Member; an average Federation colony population of 100,000 and 123 Federation colonies; to yield a total population of nine hundred thirty-five billion, five hundred ninety-two million, three hundred thousand (935,592,300,000).]
Bottom line: It's ridiculous to expect a democracy the size of the Federation to be able to count all of its votes with care and accuracy in only one Earth day.
For paper ballots yes, but an electronic ballot system should be able to yield results within a day,
And whilst the population for the UFP might be close to a trillion as you say. You forgot to account for voting age, which could halve that number.
On the continent of wild endeavour in the mountains of solace and solitude there stood the citadel of the time lords, the oldest and most mighty race in the universe looking down on the galaxies below sworn never to interfere only to watch.