Post 600, huh? Well, that leaves out the most obvious 'Justinian stays home' POD. I guess that leaves two main POD eras: a high Byzantine reconquesta under the Macedonians or (less likely) a more successful Komnenid dynasty, or an early pre-Islam POD under either Maurice or Heraclius. I'm gonna go with Maurice because A: it's probably the easiest and B: I'm a late antiquity junkie.
So for whatever reason Maurice avoids Phocas's blade. Maybe he escapes and rallies support, or maybe he just avoids the whole thing by not stationing his forces north of the Danube for the winter. He cuts his losses and continues to let the Exarchates fall away; whether or not he lets one of his sons become the western emperor is immaterial; by the end of the eighth century there's really nothing left. Ravenna has fallen as per OTL and Africa is independent and currently being swallowed up by the Berbers. Maurice and his successors keep Dalmatia though because it's a useful buffer. The Pope looks to the Franks as per OTL, but the lack of Iconoclasm is probably going to butterfly a Germanic western emperor, especially if Maurice's offspring are still clinging to life in Syracuse or Carthage when it happens. On the other hand, you're still going to have issues between the east and west churches thanks to the Monophysites and the east's continued waffling and attempted compromise. Eventually there will either be a definitive split between east and west or an Egyptian rebellion (also likely!). But the East does have four of the five Patriarchs and the Holy Land, so it has rather powerful cards in its favor.
Byzantium won't be the focalpoint of European culture due to religious and political scisims; to do that you'd need an earlier 'no Justinian' POD so that the fiction of Byzantine hegemony is maintained for longer. But hopefully it won't be as bad as OTL. The east might not even ever be really delegitimized in the eyes of the west.
With the Sassanid war cancelled though, it's likely the next conflict with them will be just another border war--and probably won't be until Maurice passes, at least. This leaves the empire more than capable of dealing with the continued Slavic threat and holding the Danube frontier.
Basically, the empire survives. It's touch and go for a bit, but with a string of mildly competent emperors it manages to pull itself out of the hole Justinian put them in. An Arab invasion, if it happens, is repulsed and after a few key defeats against both empires, the movement fractures and dissolves. So, rather than a typical Byzantine Empire we have a post-classical Roman Empire that keeps many of its traditions as well as its trans-Mediterranean urban culture. Many things will still progress similarly to OTL: Orthodox art evolves recognizably, the church continues to increase its dominance, Diocletian's reforms reach their endgame (though, perhaps not to the same extent). The empire is a rich, urban, centralized and autocratic state with a highly effective and adaptative military. Assuming it survives the challenges of the seventh century and doesn't do anything stupid (we might need a string of emperors to disavow the idea of retaking the west), it is in a perfect place to dominate the majority of the Middle Ages.
After that is when things become more...complicated. No Ottomans means no immediate need to find an alternative to the Silk Road--but someone's eventually going to find the Americas and pillage it anyway, it just might be a bit later. An alternative age of colonization is tricky for Byzantium. It's a Medditeranean power, that they tended to decline in relevance once the Atlantic was open season. They could, as so many TLs on here have decided, try their hand at the Indian Ocean, but they're still not super well placed for that.
Byzantium risks turning into an ATL Ottoman state, slowly losing ground to the Europeans as the west colonizes and industralizes, instead becoming increasingly 'backwards' and irrelevant as it continues its pointless border wars with Iran. Eventually nationalist feelings will threaten to overwhelm the state and it will be hard-pressed to keep itself together. The European powers allow it to cling to life because A: it's the Roman Empire (and they have more respect for it in this TL) and B: like the Ottomans, it's a useful counterweight to the ambitions of the other European states.
But there is always the chance that something else happens, and honestly in order to keep those borders to together the modern day, I would argue it almost has to happen: revolution. Either a swing towards a constitutional monarchy (could be fun) or a full on communist revolution. Seeing as how it's an Eastern European monarchy however, I doubt it'll be peaceful.
(Interestingly though, the Senate probably isn't reduced to irrelevance by reforms in the tenth century, so they might be a starting place for republicanism--as an idea if nothing else, since it IS still a rich old man club.)
So, onto the questions.
1: How powerful is it on the world stage? Assuming it's not a mortibund Ottoman counterpart, it's a powerful regional state that's long been eclipsed by the modern-day superpowers. It essentially owns the Middle East (or co-owns, if Iran has kept up), and no one else is going to be able to touch it. It's got allies in ATL Russia and probably most of Eastern Europe (Byzantium is to Russia as our UK is to our US?). If it had colonies, it probably doesn't anymore, especially if we're assuming this world is somewhat recognizable to our own, just with different countries.
2: There's a solid chance, but far from certain. But as I've said, with a POD this late I don't see Byzantium becoming the focalpoint of Europe, and I still see a Frankish empire remaking Western Europe in its image (just without imperial legitimacy). If Greek IS the language of diplomacy, it may very well find itself replaced once the western powers start building global empires.
3: Constantinople, obviously. The other three Patriarchs remain relevant as well; Antioch, Jerusalem, and especially Alexandria as the empire's second city. Other major places include Thessalonica, Trebizond, Edessa, Damascus, and I really like the idea of Athens as a university city. Rhodes? Pergamon? Adrianople? Ephesus probably holds out longer than in OTL, but eventually the loss of its harbor is the death of it. There's probably a few smaller Egyptian metropoli. If the empire has avoided devastating invasions, then I'd expect some high urbanization. Lots of cities.
4: With POD of 602, I'm not sure you'll get a Renaissance as per OTL. There's never really a 'rediscovery' of classical texts as they've all kind of just been trickling back into Western Europe over the course of the Middle Ages. Combine that with the loss of the Muslim Golden Age, Byzantine scholars not fleeing the Ottomans, and many OTL events we take for granted not happening...if it does happen and isn't just a gradual thing, I'd say delayed.
Or maybe the West doesn't have one at all. They're blocked off from the east, and sort of on the periphery of the trade routes. Maybe a Renaissance happens in the empire instead and it trickles out from there; the empire was always very good at taking ideas from others and adapting them for their own survival.
5: Greek was always the lingua franca of the East. Like, I don't see every other language disappearing, but Greek is going to be taught in school; ESPECIALLY in the case of a Communist state or an empire that's intentionally tried to cultivate a singular nationalism.
6: I don't see why not. But as we've seen on this post, that's because anything could happen. If the empire has good luck and makes good choices, it's in a good place to become a prosperous first world country. On the other hand, it's very easy for it to become a true Sick Man of Europe (but then, I can't see it surviving).
A wealthy constitutional republic where the imperial family is the ultimate tourist trap is a fun idea, but so is a wealthy China-esque dystopia with social credit scores and heavy censorship.
(Hmm. I kind of want to do a highway map now.)
7: I think I've sort of mused on this already, but I think you're going to see a Christianity that doesn't ever formally break. There's going to be a difference between east and west, but even with the Monophysites (who I think the empire will eventually attempt to extinguish in order to please the west), there's not going to be anything that puts blood in the water. Iconoclasm never happens, nor does the Fourth Crusade, nor do the other Patriarchs fall. The Pope might pretend to be first among equals in the west, but he still addresses his eastern counterparts with the respect they deserve.
