Why is the Emperor Beyond The Sea in The Chronicles of Narnia named that? What sea is he beyond, and why is he beyond it? Is this implying that the Emperor Beyond the Sea has a country that's kind of analogous to Aslan's Country? If so, why are they separate countries?
Emperor Beyond The Sea represents Yahweh, Old Testament God, The Father. Also, God of the Jews. Aslan represents Jesus and New Testament. Emperor indeed has a separate country (Israel, if you will) with his own people. You might say that he is beyond the Sea of Understanding or Knowledge; nobody can know his will or anything about him. In a way, Emperor is a hero of another story, a story that is almost finished. Being human god hybrid, Jesus is much closer and easier to understand than Yahweh; so it goes with Emperor and Aslan. Emperor is idea, far away from people's eye, unknown and unknowable. Aslan is there and approachable.
That's my interpretation at least.
The Emperor-Beyond-the-Sea is obviously one expression of God as he is in the context of Narnia. So Narnia is another world that God (or the Emperor-beyond-the-sea as He is called here) is lord of, where He has made Himself known to the inhabitants and sent part of Himself in the form of Aslan for their redemption. Judging from your question, I assume you have not read all seven of the Narnian chronicles- the Voyage of the Dawn Treader and the Last Battle clarify this point very well. In the former book, King Caspian and his crew take the boat through the Eastern Ocean (that's the sea) searching for the end of the world and some other interesting items. Here comes the kicker, beyond the unexplored Eastern Ocean and the end of the world lies Aslan's Country, where the Emperor-beyond-the-sea resides. Aslan's country and the land of the Emperor-beyond-the-sea are one and the same, it's just that we the readers know Aslan better and will more readily comprehend this otherworldly place if we associate it with him and not the more mysterious Emperor, hence the name and the confusion there.