They are 99.99999% compatible with an Eastern Catholic Divine Liturgy as well
In an Eastern Catholic Divine Liturgy, we also pray for the Pope, the Patriarch, Metropolitan and Bishop of the Eparchy. In a Roman Mass usually its just the Pope and the diocesan Bishop. But of course the Pope already inherently is the Patriarch of the Roman Church as well as the Supreme Pontiff. That distinction is clear in the East because its different persons.
Eastern Catholics pray for Orthodox Christians and Orthodox faith as well. Of course if you ask the Orthodox, some of them will say Eastern Catholics are not Orthodox Christians. ** Eastern Catholics see themselves as Orthodox Christians in communion with Rome.** Although in my Divine Liturgy book, the term “orthodox” is not capitalized.
Although most Orthodox will not seek Communion in a Catholic Church for many reasons, even in cases of necessity.
This (what I bolded in your post) reminds me of what I learned about the Union of Brest, and St. Josaphat Kuncevyc. He was an Orthodox Christian who wanted to come in communion with Rome.
So, if I understand you correctly, you do pray for those Orthodox Christians who are not in communion with Rome (who reject the Pope), correct?
Regarding Roman Catholic Mass, I learned that it is offered for all inhabitants of the parish, including non-Catholics, and it is offered for people everywhere, again including non-Catholics.
There’s also the Divine Mercy Novena, the devotion that Jesus Christ gave to St. Faustina Kowalska, the Polish/Lithuanian nun in the early 20th century. In the Divine Mercy Novena, we specifically pray for “those who separated themselves from the Church” (people who were called heretics and schismatics in earlier times).
Regarding the practice of EO not in communion with Rome, I remember that at DL at ROCOR, they would pray “for Orthodox Christians everywhere, for Orthodox Christians (parishioners) present at this service, and for those who are missing this service for honorable reasons”. Something like this, but I’m paraphrasing, trying to recall as best as I can. Thus, I have the impression that they at ROCOR (and now the Russian EOC, because ROCOR reunited with the Moscow Patriarchate in 2006) do not pray for non-Orthodox people, but I welcome being corrected if I’m wrong on this.
However at the Western Rite EO DL of the Clerks Secular of Saint Basil, they do specifically pray by name for the EO Patriarchs, AND for the Pope. They list a bunch of Patriarchs, including the Russian, the EP of Constantinople, the Alexandrian, Antiochian, Romanian, and so on, and they do include the Pope of Rome in this list.