Are they all subject to the guidance and discipline of those in Rome?
driftdive
Fair question and a good one. Yes, there are under the guidance and discipline of Rome, but there is a separate Canon Law that describes the role the Pope plays in the relationship between Rome and the Eastern Catholic Churches. Here is link for the 1990 Code of Canon Law, and note that his is for the Eastern Catholic Churches.
http://www.jgray.org/codes/cceo90eng.html
Since the Liturgies of the Eastern Catholic Churches are of the same Apostolic Tradition as the Roman Rite and all stem from the ancient Churches of Alexandria and Antioch, which along with Rome were recognized as the Primatial Sees of the the Church [see Canon 6 of the Council of Nicea], the Pope has no authority to change the Liturgical Traditon and devotional traditon of the Eastern Churches. In addition, the Pope usually does not appoint Bishops in the Eastern Catholic Church, which he does for Roman Rite Bishops. Rather, Synods of the various Eastern Catholic Churches, made up of Eastern Bishops and Priests, select their Bishops and then that Bishop’s name is sent to Rome and the Pope Confirms the choice.
I hope this helps.