AFAIK, the king also goes through the royal doors for communion and stands before the altar, communing as the priests.
in the ancient church yes. im not sure about modern times.
Are there any kings left?
Orthodox ones? Greece had one until 1967, though I've never heard of him doing that.
I don't think the Church of Greece ever afforded that privilege to their kings.
Although King Constantine II fled Greece in December, 1967, after a failed and amateurish coupe attempt against "The Colonels" regiem, I think the Greek Monarchy was abolished in 1974 through a plebiscite, when Prime Minister Constantine Karamanlis assured the King he would handle everything, so that he wouldn't have to campaign, but Karamanlis actually worked covertly in support of the abolition of it.
I think the poster is referring to privileges of the Tsars within the Church of Russia. I don't know about Romania, Serbia (Yugoslavia), Bulgaria, or, Georgia, either.