I'd like to add to this
1) Zagros - you're entirely right regarding Constantinople, but there is Georgia to consider. They could enter from the north, and whilst I doubt they could conquer Persia (go them for trying though), they could set up a strong frontier, or Roman-allied buffer state in the north west that Georgia and Consantinople can support to prevent a Persia secure enough to march west. Short of a strong ally helping Persia break that client.
Honestly it is 100% plausible for Georgia to conquor Iran and become the new Persian polity. At least until the Muslims throw them out.
The Safavids conquered Iran from a power base right where the Georgians are which was both smaller and not held for nearly as long. They too were foreign invaders (Turks) with a different religion (Shia) but were able to manage the creation of a new Persian polity. The differences are more extreme, however, as Turkic peoples had ruled in Iran since the Seljuks and the difference between Shia and Sunni is less than Orthodox Christian and Sunni, but there is some historical precedent that as state geographically situated where Georgia is could conquor Iran.
It is unlikely to occur, however, for the reasons listed above but it would not be difficult for the Georgians to break away the Azeris as far south as Tabriz from the Persians and act as a suitable threat to the Iranian Plateau to keep Persia either A) Cowed or B) Similar to French Revanchism for the return of Alsace-Lorraine.
2) Caliph. I'm curious as to whether you think the Caliph could become a Civil office inside the Empire (note: not explicitly an Imperial Office). Perhaps as a method to assist in keeping the peace if the Romans did take a lot of Muslim territory. A Caliph in Constantinople responsible for keeping the peace could be a savvy move.
I think a Caliph would not be considered legitimate if it was not an independent entity. However, the Mamluk Sultans
were able to do that for the Abbasid Caliphs who they kept as puppets in Cairo. But Sunnis keeping Sunni puppet Caliphs would be different from a Caliph under control of an Orthodox Christian Roman Emperor.
Your proposal would work better not so much for a Caliph but more along the lines of an analogue to the OTL Ottoman offices for the religious minorities in the Empire as part of the
millet system. The Hakham Bashi of Constantinople was considered the overall religious head of the Jews within the empire, the Ecumenical Patriach the head of Orthodox Christians, and the Armenian Patriarch of Constantinople the head of Oriental Christians. Each one had broad reaching ability to legislate for justice while Muslims were overseen by the Grand Mufti of Constantinople who served at the pleasure of the Ottoman Emperor as Caliph.
The Romans could have a Grand Mufti fill the role you describe and probably already do somewhere but the office would need to be formalized to become part of the Roman administrative apparatus. The only way to ensure peaceable relations with minority religious communities, per historic precedent, is to not interfere with their religious law (Christian Canon, Islamic Sharia, Jewish Halakha, etc.) and let them legislate for themselves. People just don't like having values imposed on them. Personally given the antagonism the Romans have towards their Islamic populations I don't think they would be willing to formally allow that sort of legal concession and would prefer to keep everything state controlled no matter how much the Muslims may complain about their religious law being violated. Orthodox communities will likely complain just as much as Catholics did IOTL when canon law was violated but that won't stop the Romans from trying to keep state control of the courts. If they won't grant legal autonomy to the Greek Orthodox Church, they won't grant it to the Muslims.
3) Marib. This needs to be rebuilt. I wonder how effective a swathe of monsoon dams would be for Roman rule in Arabia. On one hand it'd help the locals and create new population centres, but would it cause immigration from elsewhere in the Empire that could lead to conflict?
Funnily enough the dam wasn't rebuilt until the 1980s I think, and by one of the tribes forced to Emigrate due to the collapse of the Yemeni water system. Again I think.
It'd be possible to rebuild it and increase local agricultural potential and this economic value of the territory but the social outcome of it would be more up to B444's narrative desire. Large migrations of Heartlanders to the region is possible but right now would be unlikely, as the depopulated Levant is much closer to home.
Something that could be more interesting to the Romans is to leverage their high capital and soft power resources to rebuild the dam, and other pieces of local infrastructure, as a sort of foreign aid package/bribe. There is potential to retain ownership of said infrastructure for a joint-stock company of some sort in Constantinople similar to the Suez Canal Company of OTL which could pull incomes from Arabia to Constantinople. Foreign ownership of a vital piece of local infrastructure could force locals to tow the Roman line politically, assuming they even bother letting the Romans keep ownership and don't just nationalize the thing. Threat of Roman reprisal would prevent it, as was the case for most foreign-owned assets that were nationalized IOTL, but only so long as that threat is high and the need to nationalize low. It could also breed unnecessary antagonism so the pros and cons would have to be weighted by whoever decides to take up the challenge of rebuilding the extensive and sophisticated network of dams and irrigation that characterized pre-Islamic Yemen.