Hey Talmid,
The Antiochian Patriarchate is one of the original Patriarchates, and our Patriarch succeeds from the Chair of Peter, (is that the right phrasing?) He now resides in Damascus though, on the Street Straight actually, because Antioch has been decimated by earthquakes and things. The Antiochian Church is one of the main three in America, along with Greek and Russian, although there is an effort in process to unite the Church in America in one diocese. The problem in the USA occurred because of the Russian revolution in 1917 and the persecution of the Church in Russia. They had been in charge of things here until then, but because of the disarray, each church turned toward their national church, and things have remained that way since. It will eventually become an autonomous American Church, but things move SLOWLY in Orthodoxy.
I am not that well versed in all the ancient Churches, but I agree on your assessment of the Coptic, Assyrian and Ethiopian. They are Eastern, but there is some point of disagreement, and I think they are not in communion with the main Churches, and I THINK it has to do with monophytism. I know the ancient Iranian Church was Nestorian, but I don't know if any parts of that still exist.