And then there are times in my OCA parish where the Doors will remain closed, but priest will open the 'curtains'. First, what are the 'curtains' called and second, do the Greeks and Antiochians have them?
It's called a curtain, and, yes, Greeks and Antiochians have them. There's been a trend away from them in recent construction, but they've been standard for a long time. Curtains on the ciborium is the older practice, but we haven't built ciboria in our churches for probably 1,000 years.
I've actually seen photos of some Orthodox Churches with Ciborium even today. I don't know who new/old those parishes are but I've even seen photos of Russian Churches that have them.
Originally the templon had a curtain all the way around in some instances, and in some it was the ciborium with the curtain.
Since in our church, the ciborium turned into the tabernacle (the item on the altar that holds the eucharist) and can't hold a curtain, and since the templon turned into a more solid iconostasis, the only real place for the curtain was the Royal Doors.
This is a good reconstruction of the way Hagia Sophia probably was with the curtain around the templon:http://youtu.be/BFjNLX64r9M
I think there were also curtains on the ciborium, or at least around the altar table because apparently one of the Patriarchs allowed a politician to hide under there when the authorities were seeking to execute him. The Patriarch gave a sermon that Sunday against the behavior this politician exhibited (can't remember what it was) and during the sermon walked over to the altar and pulled back the curtain, revealing the man hiding beneath it. The man was eventually given over to the authorities and judged.