The only think that damn hard to explain is them being members as far back as the late 23rd Century, if not further, and it being such a shock in "The Host" that Trills are joined with symbionts. (Then again, considering DS9 basically ignored everything about that episode except for that basic premise, maybe we should too).
We've seen UFP members keeping customs and facts about their culture from the Federation before; for instance, the Ardanans' system of apartheid with the Stratos elite keeping the Troglytes oppressed, something that would probably have precluded membership if it had been known. (I always figured the UFP was willing to rush Ardana's membership without looking too closely because they wanted its zienite badly.) The Federation is a looser union than something like the USA or even the European Union, simply because the worlds are so far apart. Internal matters are generally kept internal.
This one's a bit trickier. The whole storyline of "Let He Who is Without Sin..." seems to put Risa as a member, but it's hard to reconcile that with Picard seemingly being unfamiliar with the planet and Riker explaining it to him in "Captain's Holiday".
Not really. I'm with Sci
on this one. Do you know the name and customs of every one of the nearly 200 sovereign countries that currently exist on Earth? Did you even know there were that many? And that's just one planet's worth of sovereign states. In an interstellar union of hundreds of entire worlds, it would easily be possible for a citizen of one member planet to go through an entire lifetime without learning a thing about many of its other member planets. Sure, Picard's a well-traveled fellow, but he's spent most of his life discovering or exploring worlds beyond the Federation's borders.
Although, granted, given that Picard was more of a rake in his youth, it is a little unlikely that he was unaware of the world that was later established as the go-to pleasure planet of the quadrant even as far back as the 22nd century.
USS Einstein wrote:
I.E. 300 years of immigration mean there are probably thousands of alien species, even from beyond the Federation, living on Earth.
Good point. One thing that ST and its fans all too often forget is that species identity doesn't automatically correspond to political or cultural identity. A member of the Andorian species born and raised on Earth would be a Terran by nationality and culture, and a human born and raised on Vulcan would be a Vulcan citizen. But the shows always simplify things for the audience and make characters act stereotypically based on their species. Worf was always the ultimate Klingon even though he'd been raised by humans and spent most of his life on Earth. Although we did get some rare exceptions, like Jono/Jeremiah Rossa from "Suddenly Human" and Rugal from "Cardassians." But these were always treated as anomalies and the characters' default reaction was that they should be with "their own kind" instead.