As far as I know, monophysites were the predominant type of Christianity in a lot of the eastern provinces of Byzantium, so that's probably the version he would have picked up and started spreading. I doubt he would do political unification though.
I think the arab invasions would be more like the germanics than the mongols, because without Islam there is nothing to unify the Arab tribes. IMO they will likely take over Syria and Mesopotamia, possibly Egypt too, before they get pulled into the Byzantine and Persian orbits.
You're right, they won't abandon Italy altogether, but it certainly won't be their main focus. In the early middle ages, Egypt and Syria were much richer than Italy, so they would probably put more effort into Egypt and Syria. Plus, the church in Rome will still be irritated at Constantinople for multiple reasons: 1) being the chief bishop of the church, they will be a major target for Constantinople to control, but far enough away to resist, so the emperor will be inclined to try to elevate the patriarch of Constantinople (who he can control much more easily) above the Pope, thus irritating the papacy, and 2) in real life, Constantinople was always trying to find some compromise with the monophysites who dominated their eastern provinces, and this compromising of chalcedonianism always irritated Rome. OTL, Constantinople stopped after the Islam invasions rid them of their monophysite population. Since there is no Islam invasion in TTL, Constantinople will still be trying to compromise with the monophysites, still to the irritation of Rome.
So maybe the Pope allies with the Lombards and Franks against Byzantium, and since Italy was significantly richer than Francia and Germania, so the Franks will be more concentrated on Italy than the Byzantines. Maybe the Pope still crowns Charlemagne Emperor, for defending him from Byzantium.
there still could be muhamed was one of the people sure religion helped but there is a timeline were abu bakar with khalid or any other influential early islamic figure could have united the arabs like chingis did with , so i guess in your timeline it would stil be successfull arab invasions this an acceptable point of view but do remember than they could still get crushed since there is no unity diferent arab tribes or cofedracies attack later and in diferent times giving the romans more breathing room.
this would make kinda of sense your abandon italy idea for if the arabs invade and take syria and egypt then italy migth be lost but then again the arabs are not as big as the otl .
for the rest it falls in to anachonist points of views of the early medival period
1) the emperor would never did and nor would they will evelate the patriach of constantinople to the pope they did however reduce the pope authority (the one he was gaining) back to equeal of that of the others.
"in real life, Constantinople was always trying to find some compromise with the monophysites who dominated their eastern provinces, and this compromising of chalcedonianism always irritated Rome. OTL, Constantinople stopped after the Islam invasions rid them of their monophysite population. Since there is no Islam invasion in TTL, Constantinople will still be trying to compromise with the monophysites, still to the irritation of Rome."
this is frankly bs , heraclius was the only one trying this , justinian and his predeccesor actively promoted chalecedonian views ,justin II actively perescuted the heretics ,tiberuis II just tolared them and the same was with maurice.
Heraclius was a war hero in the eyes of the people and even then his compromise was not fully accepted , if the arabs invade despite not been as total heraclius repupation will fall and his compromise will die with him, as they lost the provinces and what good is a compromise if the other side belongs to the faith.
If how ever we delay the arab invasions till the death of heraclius in the early 640s , then there is the problem that gives rome more time to recover and kill any arab invasion with no other power left the byzantine could then impose really whatever they want but then again they dont have to apease no one so you could get rid of the compromise that both monophisyes and chalecodonians hated and continue maruice tolerance of them.
"So maybe the Pope allies with the Lombards and Franks against Byzantium, and since Italy was significantly richer than Francia and Germania, so the Franks will be more concentrated on Italy than the Byzantines. Maybe the Pope still crowns Charlemagne Emperor, for defending him from Byzantium."
any pope that allies with the lombars is as sure as dead , also the frankish realms wont be able to do much see this video for information
If they could by the late 8th century then a lot of things could and can change by that point
That the franks really won't be able to do much .
Depends on how the Byzantine go about in he 7th And 8th centuries
some safavid sources state that some isolated villages in Mesopotamia still had arabian polytheistic majority in the 17th century, so it does show that the arabian polytheists did have *some* staying power.
quite unlikely unless they are refering to remote areas around najd area there it makes sense for the imporant parts ie the coast it was 2 centuries at max so the idea of them becoming chirstian in less than one is lauguable .