Southern Italy stays Orthodox, maybe a stronger Byzantine Empire leads to a Orthodox conversion of Lombards instead a Catholic one ?
Again, thinking in terms of Catholic vs. Orthodox makes little sense before the classical Middle Ages.
Byzantines never really bothered at the cultural and doctrinal differences with Latin church
es, as long the ones that were under their domination acknowledged imperial dominance.
They become vassals of Byzantium
Vassality as we generally understand it, was appearing by the mid VIIth century in Francia : Lombards structures were significantly different, even if related (would it be, as in most of Merovingian Francia, the distinction between benefici and title).
and fight against the Catholic Franks, their archenemies ?
Which they were not. After the Lombard takeover of Italy, the
Regnum Francorum was essentially driven first on an unification trend, and then mostly against peripheries (especially Frisians and Saxons).
What would happen to Pope´s Rome in such a case ?
At this point, the predominant pontifical position among Latin church simply didn't exist yet : that's more a byproduct of the Carolingian era.
I'd expect a longer lasting Exerchate of Ravenna, forcing Lombards back north of Po (roughly), to be significantly autonomous from Constantinople, making the Pope a political factor within Byzantine Italy.
Some sort of intermediary between western kingdoms churches (which would probably keep a "national" character, as in having the high clergy and the king as its head, as a continuity of Late Imperial relation with Christian churches) and Byzantium, maybe, with a strong symbolical significance as to both mark Byzantine influence in Europe but as well distinctiveness of the latter.
Southern Sweden did have some traces of Greek Christianity before the Roman one became too strong.
As far as I know, tough, the first missionaries attempt recorded are Frankish-supported, around the late VIIIth/early IXth century. Onwards from this, Latin pressure was already quite strong.
That said, traces of Greek Christianity there would be far from unthinkable, but I just don't remember about them : do you have some links?
Unless that you were mentioning Greek Christian art and artefacts found there? In this case (but that's so different that I don't think it was what you meant), it's more clues for trade than actual religious presence.
Anyway, could not the Emperor in Constantinople push for reducing the Pope to Patriarch of Rome, and installing a couple more Patriarchs in the West, say in Avignon and Regensburg, to limit the Pope.
By the time the pontifical dominance on Latin churches is a thing, a Byzantine reconquest of Italy in such hostile matter would be doomed to fail at short term.