- Yes, Eastern and Oriental Catholic laity pray the Jesus Prayer. You’ll find it also has a growing devotion in the West as well. Some from the Syriac traditions also pray the Jesus Prayer, but it’s not really part of the Syriac tradition per se. The rosary is also prayed in the East, but there’s not as much of an emphasis on it since it’s a Western devotion.
- There are a number of reasons the East focuses on "have mercy on me… One reason is humility. If you read Eastern mystics, they emphasize regarding oneself as the greatest of sinners, and even regarding others as angelic compared to oneself. It’s thought to be presumptuous to consider others needing God’s mercy more than oneself.
A second reason is the Eastern attitude of “save yourself, save the world” (if I may boil it down like that). A favorite quote from St. Seraphim of Sarov is “Acquire the Holy Spirit and thousands around you will be saved.” This is somewhat akin to the saying attributed to St. Francis of Assisi: “Preach the gospel at all times. When necessary, use words.” In the Eastern mentality we don’t focus on ourselves as an end, but rather we believe that once we acquire the gift of the Holy Spirit (which only comes as a result of our purification from sin and as a gift/grace from God), then we become more effective instruments for evangelization.
Finally, it’s important to bear in mind that the formula for the Jesus Prayer isn’t as fixed as the “Hail Mary.” Variations abound, and one is free to vary the prayer to suit one’s own spiritual journey. Met. Kallistos Ware, for example, speaks of praying the Jesus Prayer as: “Lord Jesus Christ, Son of God, have mercy on us.” St. Seraphim (and others) have also recommended offering the Jesus Prayer for others in this way, “Lord Jesus Christ, Son of God, have mercy on your servant/handmaid [Name].” I find the latter form a wonderful way to pray for the folks who ask me to pray for them.
- Why the focus on the Jesus Prayer? One could equally ask the West, why the focus on the rosary when Jesus gave us the Our Father? The East likes to focus on the Jesus Prayer because it sums up all that we believe and all that we hope for. Just like the Hail Mary does for Western Christians, the Jesus Prayer points us to key passages and stories from the Scriptures, offering those passages to us for our meditation.
The East also has a strong “theology of the Name.” Names in Eastern culture, particularly in the Jewish culture of Jesus’ time, point to the very essence of the person - who they are. They also, in a real way (made even more real by the fact that we are invoking God the Son Himself) invoke the very presence of the person. And, as you know, where Jesus is present, the devil cannot be.
That being said, the East certainly does not neglect the Our Father any more than does the West. It is prayed at all of our Liturgical celebrations. It is recommended for our morning and evening prayers. And Eastern Christians are encouraged to say it throughout the day along with the Jesus Prayer.
I hope that’s helpful!