Welcome to the Forum. You have come to the right place.
I have been feeling very drawn to the Eastern Catholic Rites lately, particularly the Byzantine Ruthenian Catholic Rite.
Wonderful! I would stay away from using Eastern Catholic Rites and instead use the term Eastern Catholic Churches.The term Eastern Rites has a lot of historical baggage and since Vatican II the term Church is more proper as the Eastern Churches are Churches in the fullest since of the term. The Ruthenian Catholic Church follows the Byzantine rite.
I was wondering if there was a way to officially become Byzantine Catholic or if I could just start going to Byzantine Liturgy.
You don't have to be a Byzantine Catholic to attend a Byzantine Catholic Liturgy or even to become a member of the parish. Any Catholic can attend a another Catholic parish of a different ritual Church.
If after a while you feel God calling you to make the official Change of Ritual Church you can discuss that with the local Byzantine pastor. But, I would encourage you to attend a parish for a long time and pray before making that decision.
I would encourage you to speak to a local priest or your spiritual father/director about you interest in the Eastern Catholic Churches.
Also, I pray the morning and evening prayers of the Liturgy of the hours every day and is this something that is regularly practiced in the Eastern Rite or is it more of a Latin thing?
Wonderful, many Eastern Christians pray the hours of the Byzantine Church. This is something you should discuss with your spiritual father before undertaking it within the context of the Byzantine Church
If I become Byzantine Catholic can I still attend Roman Mass regularly since there is only Byzantine Liturgy on Sunday around me and I like to try to go to Mass on days during the week if I can?
Yes! We are in full Communion!
Also, are there any Byzantine Catholic religious orders for lay people, like Opus Dei, the Third Order Franciscans, Carmelites or anything like that?
This tends to be more a Latin Church practice. The Romanian Greek Catholic Monastery in California, Holy Resurrection, has a Monastic Associate program which is like a third order.
I hope this helps and welcome to the forum. Good luck on your journey East. I will be praying for you.