The UFP is a democracy. Every planet would vote for its own leadership and for its representatives in the Federation Council. Then after all that was done, everyone of voting age would also need to vote of the President. To me, that's the most straight forward system.
Every planet would have their own rules for elections, including the legal age to vote. This would depend on what age each species felt that its citizens were responsible and mature enough to make the decision on who to vote for. As I said in the previous thread, that would be completely different for long-lived species (such as the Rhaandarite) that it is for Humans.
On Earth, I think that the voting age would be 16 by the time of TNG. But again, that's just my thoughts.