Catholic Rites, what are they and how does it all work?

They are no different from the autonomous churches.Each represents its heritage and cultural influences. I care not the type of liturgical practice as long as the Dogma of faith is always preserved. The difference between Eastern Orthodox (Greek) and Byzantine Catholic , one is in communion with the Pope .
Eastern Orthodox:
The church has no central doctrinal or governance authority analogous to the Catholic pope, but the Ecumenical Patriarch of Constantinople is recognised by all as primus inter pares ("first among equals") of the bishops.

 

Yes, the way of worship can vary somewhat but all adhere to the basic doctrines of the Latin Rite. We have separate places where we worship, but one could easily go to another and say accept Holy Communion with no difficulty.

From what I know about the Eastern and Byzantine, they both have the altar behind closed gates that separate it from the congregation. But other than that the Byzantine look to the Bishop of Rome as the head Bishop, the EO does not.

We most assuredly recognize the baptisms of these other Catholic affiliated churches, in fact the Catholic Church even recognizes the baptisms of Protestant sects, provided the baptism is done in the proper manner, i.e. in the name of the Father, the Son, and the Holy Ghost.