This is a fine description of grand strategy under, say, John Tzimiskes or Nikephoros Phokas, but not for Basil II, whose intervention in the east was mostly restricted to a maintenance and consolidation of the status quo. The Fatimids handed Basil a number of perfectly good casus belli, like interfering in the ostensibly Byzantine vassal state of Aleppo and desecrating the Holy Sepulcher, but no serious military response was forthcoming. As we know, at the end of his life Basil turned his attention not to the East, but to Sicily. The claim that the Byzantines would have by their nature been more interested in the Levant/Egypt than Italy is not borne out by the facts.
Basil worked hard to crush the power and influence of the great eastern "military families." All the major usurpers (attempted and successful) during his life - Nikephoros Phokas, John Tzimiskes, Bardas Phokas, and Bardas Skleros - were military men originating from or strongly linked to Anatolia who rose to prominence on the basis of their military power and accomplishments in the east. By the time Basil was dead, the Phokades and other such military families had been greatly humbled or destroyed over a series of rebellions and land reform measures. These men represented an aggressive, eastern-focused military policy, and by the time Basil dies there's really nobody to carry the banner of that policy.
It's possible that Basil's hypothetical successor would abruptly change gears and target the Levant for expansion, but it's just as likely, if not more so, that Basil's European-oriented policy continues, in which case southern Italy and Sicily are the new military priorities of the empire. In the mid-11th century the Zirids of Tunisia turn their back on the Fatimids and shortly thereafter get mauled by the Bedouin; assuming that still happens on schedule and Sicily is Byzantine by then, you might see imperial intervention there. I don't think it's likely, but it's not the worst idea in the world.
Of course, the original question is "how would they administer it all." I can't really speak much to that, but as far as the Pope goes I will say that the mid-11th century is not such a bad time for the Byzantines to become heavily involved in Roman politics considering the regency of Henry IV. I do not, however, think the Pope is very likely to become a willing subordinate to Constantinople as in former times. More likely, you're going to see a peninsular tug-of-war between the two empires, and the popes - who never want anyone with firm control over Italy - are going to switch sides as needed to avoid being subjugated by either one. Unless the Byzantines just go hog wild and mop the floor with the Germans all over Italy, the best they are likely to get is a firm control over the south (which would be ruled in the traditional thematic style), loose influence over Rome, and maybe some independent-minded allies/clients in central/north Italy. Think late 12th century Italy (post-Legnano) except with the Greeks in the south instead of the Normans.
Actually Basil II did respond to the Fatimid invasion of Aleppo, pushing the Fatimids away from Aleppo, and then attacked Tripoli and Tartus, occupying Tartus. I would call that a serious military response. Basil just didn't want to turn that response into a full-on invasion.
Of course, it does show that Basil II wasn't really interested in expanding south or east except to establish a defensible border. But I don't think that precludes his successor from attempting the same. Once Sicily, Salerno, and Naples fall to the Byzantines, there is no immediate reason to expand further in Italy. Like you said, there would be a peninsular-tug-of-war between the Holy Romans and the Byzantines.
(However, I don't think that Byzantine rule over Sicily would be thematic. Basil II was constantly trying to decrease his reliance on thematic reserves and increase reliance on tagmata, starting a trend that would culminate in the more mercenary focused later Byzantine empire. This was bad for the not-so-capable OTL emperors who created the conditions for defeat at Manzikert, but became a good thing for the Komnenians, and would be a good thing for a hypothetical, capable Macedonian emperor succeeding Basil II.)
Once Byzantine control over southern Italy has been solidified, there is only one sensible direction to expand, and that is south into the Fatimid Caliphate--which gradually became weaker and weaker as the 11th century progressed. Aleppo was consistently a point of tension, such as in 1030 during Romanus III's reign, and war with the declining Fatimids would occur no matter what. The question is whether Basil II's successor would turn this war into an opportunity for expansion. Because of the wealth of the Byzantine treasury and size of the army after Basil II's reign, such a task would seem pretty doable for "Basil III" or "Constantine IX", and also a way to bring new wealth into the empire and more prestige and respect to the emperor who could now prove his worth after his accession.
Also Sicily was theoretically a protectorate of the Fatimids, so the conflict could start even earlier. At that point it would be a two front war, which would be a more difficult challenge for the Greeks, but they could forestall the annexation of Sicily and send the bulk of their forces into the Levant--and in 1025, the Fatimids were fighting a Bedouin rebellion, diverting their strength.
Yes, and under vastly different geopolitical circumstances. Stretching merely as far as Justinian did in the 6th century political climate dangerously overstretched the empire then.
And Justinian also had the misfortune of 1/4 of the population of the empire dying to plague, as well as the Gothic War taking 20 years. Even then, significant parts of Justinian's conquests in Italy and Africa survived for decades to centuries after his death.
The Byzantine Empire of the 11th century doesn't have anywhere near the resources and power projection the Empire of Justinian's time did.
Basil II's empire had less resources, but it was also far more self-sufficient, not relying on Egyptian grain and wealth like Justinian's empire. Basil II was also not reliant on a Belisarius but led most of his own campaigns.
Justinian had greater power projection at sea, but I think on land, the two empires are equitable. Basil II's army may actually have been larger, as well, and probably had better cavalry.
To say nothing of the fact that European states were more developed than they were then as well.
Before the printing press, more developed states tend to be easier for empires to conquer and hold, rather than the other way around. Conquering a more developed area brings new wealth and power to an empire, conquering a less developed area means it has to be built up and is probably an expense for the empire. This is why the Roman conquest of Carthage catapulted the Republic's rise, while the decline of the West caused a lapse of imperial power.
Justinian's reconquest of Italy was a drain on the empire not because the logistics of taking Italy (in fact the strait of Otranto is only 45 miles wide), but because after multiple sacks of Rome, the fall of imperial power in the west, and 20 years of war with the Goths, Italy was no longer a wealthy prize for the empire but an expensive territory to maintain.
Of course, invading wealthy polities is generally more difficult for an empire, even if holding them is easier. But in 1025, the Byzantines were stronger financially and militarily than all of their rivals, and any country in the Mediterranean region. So despite the fact that their enemies were (arguably) stronger than the enemies of the Roman Republic and early Empire, the power disparity is about the same.
That army has to be split up across several contentious frontiers.
Yes but the Byzantines were willing to recruit non-Greeks. The army would expand as the empire does.
It's easy to exploit geopolitical rivalries, but much harder to turn that into lasting conquests in places like Spain and southern France. More importantly, there's not much reason why the Byzantines would want such territory that would be much costlier to maintain control of than to just attempt to play their enemies in the region off each other.
Spain and southern France were actually very wealthy at this time. Not as much as Anatolia, southern Italy, or Egypt, but there was definitely a lot to gain by taking Cordoba, Barcelona, or Languedoc.
The Christian states of Spain were as often at war with themselves as they were with the Moors. The chances of them willingly giving up their sovereignty to an empire on the opposite side of the Mediterranean is almost 0. The chances of the Byzantines being able or willing to conquer them is also 0.
I didn't mean that the Christian states of Spain would swear fealty to the emperor (at least at first), but rather that the Byzantines would ally with them to attack the Moors. After building a center of power in Iberia (this would be at least a century after Basil II's death, and half a century after no Manzikert), they could then divide and conquer the northern Christian states, maybe making one or two of them into favored client kingdoms while conquering the others.
Logistically it is no more difficult than the Roman or Umayyad conquests of Hispania; Rome was facing a largely tribal Iberia, while the Umayyads had their center of power in Damascus and had yet to perform any naval invasions.