If, like the Visigoths, they manage to unify the Goths and the natives into one people then they can establish a decently powerful Italian Kingdom.
That said, Goths did manage to pull that relatively late in Spain, even if by the end of the VIIth century it was achieved.
While remembering that Italian society seems to have been more divisie on these matters than in Gaul or Spain (mostly due to the proximity of ERE and the strong Italo-Roman identity), I can't see Ostrogoths being able to achieve this as easily that Visigoths did IOTL.
Will the Merovingians lose power like OTL to the Prince of the Franks?
Probably not. If Romans manage to relatively easily pull off Persians and if Arab conquests are then butterflied away, it's likely you won't see the decline of mediterranean trade roads (which were weakened but still there) and especially the link between Constantinople and western Romania.
For instance, you probably won't see, or as much, the shift between the Rhone/Seine trade road with the Rheinish/North Sea trade road, happening as quickly as IOTL.
It would make Neustrian courts in better position, especially with the decline of Merovingian Francia in the mid VIIth century being moderated.
Merovingians could still loose power, but I don't think it would be facing a strong Rheinish family (let's remember that Merovingians were still more pwoerful than Peppinid propaganda made them be by the early VIIIth century)
You all think that Lombards would probably stay out Italy if Ostrogoths remain strong in proper Italy and that Lombards may just take Pannonia.
Not exactly : I think that Lombardi could still raid the hell out of Northern Italy, but that they would have an harder time establishing their territorial power outside Illyricum (and possibly North-East Italy as well) giving that Italy ITTL have more ressources to use to fend them off.
If the pressure of Avars on the Lombards is high they will try to move in anyway.
Will Avars be that powerful ITTL, when ERE fend off Persians? They did managed to put a show IOTL, mostly because ERE was less interested on what happened on its Danubian borders.
Not that they wouldn't pull off an hegemonic rule as they did IOTL on central European Slavs, but with both a stronger ERE on the region AND a stronger post-classical Italy, it would be a bit harder for them.
Not only, indeed, Avars needed byzantine gold as tribute or loot in order to keep their hegemony in one piece, but you may even see peripherical opposition to them equivalent to
Samo's Kingdom (except possibly partially Ostrogothic in origin rather than partially Frankish)
Now, Justinian plague effects are still going to play fully, disadvantaging more whoever controls Italy than an invader from Pannonia. But if we look at what happened in Francia IOTL, they didn't suffered much eventually from Saxon and Avar incursions.
I would tend to think that an Ostrogothic Italy that would manage to survive in one piece in the VIth century would have its fair chances and possibly turning Lombard as a quasi-independent duchy under their overlordship in Pannonia (not all of it, and probably taking its fair share of Illyricum) not unlike Franks did with Bavarii.
The risk is that Lombards could take advantage of some of the Ostrogothic internal fights to eventually seize North Italy at some time or maybe Italy develops into a scenario of a Lombardic North vs an Ostrogothic Centre/South.
That's admittedly possible, but for aformentioned reasons, clearly less likely to happen than IOTL.
A strong Ostrogothic power in Rome which would not allow Frankish interference might lead to a Papacy with no secular power and no Papal states.
Popes didn't need Franks to have temporal power : the Ducatus Romanus was largely autonomous from ERE by the VIIth century. Now, there's indeed the point to be made that an Ostrogothic kingdom wouldn't allow a quasi-independent pontifical power in Italy.
That said, you're largely underestimating the secular power of Papacy within Italy, especially as Ostrogothic political power was declining : because that secular power was largely symbolical doesn't mean it wasn't real when it came to municipal influence and on the Latium in general.
If it is deprived from the relevance it achieved IOTL later thanks to the Franks, other powerful bishoprics could raise in the West (Cologne?, Tarragona?) that could eventually challenge the alleged leadership of the bishopric of Rome.
You're confusing two things : Papacy as having a quasi-monopoly on Christiendom which was achieved IOTL, and Papacy having a significant (and unchallenged) dominant role over Christiendom.
Long story short, not only papacy was never only about the bishopry of Rome, but all Western Christiendom structures depended (even if from a symbolic level) from acceptance of Roman legitimacy, if formally.
Even in the Early Middle Ages, when church policies outside Rome were largely managed by national councils (most famous being Toledo's, but you have such for every entity), these national councils not only never challenged Roman legitimacy, but abided to it in order to get legitimized as well.
The Visigothic Kingdom was neither stable nor strong enough to challenge the Franks by its end in 711, but if they would have survived (no Arab invasion) they could have probably interferred in some Frankish policies, i.e. the dominion of Aquitaine or the control of Provence.
We discussed it in this thread : the consensus was that the Visigothic Kingdom went trough an important political crisis, and would probably rather be the target of Frankish interventionism than the reverse.
Basically : while Francia underwent an unifying trend, Gothia went the reverse road with a tradition of sub-kingdoms or separate kingdoms (as Paulus' kingdom) reagularily popping out.
Eventually, while there's a whole record of Frankish raids and interventions (as in 632, when Frankish help was decisive into the succession crisis of the time), when Goths were forced to see their northern provinces going more autonom (Toledo Councils generally had exemptions for provinces such as the province of Gaul).
- Do you all think that, after an eventual decline of the ERE influence in the West, some of the Germanic Kings (Frank, (Visi-/Ostro-)gothic, Saxon...) would have tried to impose his overlordship over the rest and to acquire some analogue title to Emperor/Great King, thus finishing the symbolic overlordship of Constantinople there? (I am talking about two centuries or so after Justinian).
It's hard to answer, but I'd go with no. IOTL the rupture between post-imperial western Romania and imperial eastern Romania was due to relatively specific conditions.
Namely :
1) Justinian conquests (that got rid of "buffer" regions in Italy and Africa, making the distinction between Romano-Barbarian kingdoms and Roman Empire more clear
2) The long Romano-Persian wars that weakened the commercial and political communication between East and West
3) Growing ritual differenciation between East and West (essentially due to how Byzantium abided by more authoritarian take on dogmatic definition, depending on how the war went with Persians and Arabs)
4) The Arab conquests that achieved most of its rupture.
Without all of this, and with a stronger post-classical Mediterranean continuum, I wouldn't see pretention to imperium over Christiendom (because it was what Carolingian imperium was all about) being seriously challenged, at least not that quickly.
Maybe later, but for what interests us? I don't think so.
That said, you could have pretention to hegemony in western Romania, without the need of claiming an universal imperium.