Thank you for the replies, everyone.
Although Basil was a legitimate dynast, and a competent plotter, as evidenced by his deposition of Basil the Parakoimomenos, I'd imagine that he would have some difficulty deposing Tzimiskes. John's prestige would be high, especially amongst the army, due to his string of victories. The memory of Romanus Lekapenos would still be alive at this point, so I imagine that no one would be surprised to find John carrying on his reign beyond the necessary point. As a best case scenario, Basil, and John, may be able to come to an understanding, as Tzimiskes doesn't have any sons, and his maternal relations, the Phokades, loathe him.
If they can't, and civil war breaks out anyway, I'd imagine that Basil wouldn't be in personal command, as he doesn't seem to have been an excellent commander himself, although Bardas Phokas would defect to his side, and Bardas Skleros might, as he had tried to launch a coup before. I'd imagine that Basil the Parakoimomenos was relatively happy under John, as he was able to govern in Constantinople, whilst John was in the east. As IOTL though, Basil II shall most probably depose him. If Basil wins, I would predict another round of civil war, as both Phokas, and Skleros, desire the diadem.
I can't imagine that Bulgaria would be able to break away, although she was already in revolt. A competent general, rather than the inexperienced Basil, might be able to crush the rebellion in its infancy. Even if not, should a competent general be victorious in the civil war(s), they should be able to conquer Bulgaria rather easily, as long as the eastern frontier is quiet. IOTL, Basil's conquest of Bulgaria seems to have been a parade of mediocrity, as after the cull of the civil wars, Byzantium suffered a dearth of competent generals. It is rather revealing that, when Nikephoros Ouranos was victorious at Spercheios in 996, there was many a celebration in the empire, as the Byzantines were used to defeat by this point.
From the biography of Basil I've read (Basil II and the Governance of Empire, which is an excellent book although the 300 pages of source analysis are a bit dull), he doesn't seem to have been overly harsh with the aristocracy, but instead cultivated the image of an all-powerful autocrat. For example, he allowed members of the Phokades family back into the military, despite the grasping for the throne by many of the house. The example of Eustathios Maleinos can be explained as putting an over-mighty subject in has place, as there is suggestion that he was using ostentatious displays of wealth as a threat to the emperor not to mistreat the Anatolian magnates.
In regards to the Empire becoming less focused upon Constantinople, I would argue that that is a certainty so long as the Empire is able to maintain her hold over the newly acquired lands in the east. The increase in wealth, and population, about to be brought about by the coming economic boom, and the reconquest of other urban centres, particularly Antioch, although it had undergone a decline from the glory days of Antiquity, shall turn Constantinople from the sole large city of the empire, into the greatest of a group.
The idea of a more religiously tolerant empire is certainly intriguing. I'd imagine that Muslims would be treated better than the Jews, as, although they were protected, the Jews suffered from the general apathy with which Constantinople reacted to disobedience of the legal code in the provinces. Consequently, they were subject to illegal pogroms, such as when St. Nikon of Sparta drove the Jews out of Sparta. The Muslims, making up at least a very sizable minority in the east, would be immune to this sort of treatment. How the empire might deal with the increase in the amount of monophysites over the long term is also interesting. In particular, shall it become imperial policy once more to attempt to reunite the Church, as attempts only ceased in the 7th century IOTL, due to the loss of the monophysite provinces?
IOTL, during the campaigns in Upper Mesopotamia by John Tzimiskes, there were riots in Baghdad as people felt that Adud al-Daula wasn't doing enough to stop Byzantine expansion, and keep Baghdad safe. Should the Byzantines continue to be successful, it might make Baghdad even more difficult to govern.