At the end of the play, the mane six are standing in pairs as they sing: Applejack/Fluttershy, Twilight/Rarity, and Pinkie/Rainbow Dash. There doesn't seem to be much meaning to this arrangement, until you remember that Fluttershy was originally going to be an earth pony and Pinkie Pie a pegasus.
The three tribes in the play correspond with the three classes described in Plato's Republic: the Earth Ponies are the Producers, who are laborers and farmers, the Pegusi are the Auxiliaries or warrior class, and the Unicorns are the Guardians, who serve as the philosopher kings.
"The Earth is round. There is no up or down." Does this mean that their planet is called Earth? And that Smart Cookie proposed giving the new earth pony nation the same name as their own planet?
That could be a writing error, or perhaps an error on Pinkie Pie's part. "Earth" might not be the name of the planet, but Pinkie was thinking of her next line. Contrived, but it's Pinkie...
Princess Platinum inherited her position and Commander Hurricane probably gained theirs through Klingon Promotion. Chancellor Puddinghead, however, got elected as a leader somehow. Think about that for a while.
Maybe they're taking pot shots at real-life elected government officials. Or they know Puddinghead is a lot smarter than s/he looks and acts. Or quite simply, Puddinghead was elected for his/her hatred of the other two races, and was seen as somepony who could do something about them. There are a number of reasons why Puddinghead was elected. Puddinghead is creative (in a weird way).
Despite Equestria being formed by the unicorns, pegasi and earth ponies before they were around, Celestia and Luna are still featured on the first flag of Equestria.
Most likely the ponies just used a modern flag or one from when Celestia and Luna took the throne after all we don't know the amount of time between hearts warming and when the royal sisters took down Discord.
The Journal of Two Sisters elaborates that the ponies founded Equestria, gave sovereignty to Celestia and Luna before they were usurped by Discord.