If you start off being baptised into the Greek version of the Orthodox faith then are you always known after that as Greek Orthodox or just plain Orthodox??
Someone might jump on me for saying this, but there is no such thing as a Greek Orthodox Christian -- there are just Orthodox Christians who are under the authority of a bishop of the Church of Greece, Cyprus, Constantinople, &c.
I am under the authority of the Archbishop of the Greek Archdiocese of Australia, but I never refer to myself as "Greek Orthodox" and would quite happily commune in a Russian or Serbian or Bulgarian Church.
Joining that question is this one as well.... Do you have to have a Greek Saint (and not any other one)
Any name of any saint of an Orthodox Church is acceptable. That's how the Russians ended up with names like Nikolai and Fyodor, after all.
and last one.... Do you come under a certain jurisdiction all your life or does that change depending on where you move to and what type of Orthodox church you go to?? Like if you start out Greek baptised then go to a Russian one even though you can attend any Orthodox church... do the Russians look at you and think yeah but your Greek Orthodox because that's where you were baptised??
This one is more complicated. While you will not be refused communion in a Russian Church if you were baptised in a Greek Church, it must be kept in mind that in the past it was not possible to jump from jurisdiction to jurisdiction without actually physically leaving the territory of your particular bishop -- eg: crossing the border from Constantinople's territory into Bulgaria's territory. You should always keep this in mind when considering issues of jurisdiction and the authority of the bishop.
