Very difficult. The very act of winning back such large amounts of territory exhausted the empire in terms of both manpower and treasure. In short, too much was won back too quickly. If Justinian had contented himself with the reconquest of North Africa and left Italy to be regained over the next century or so by his successors, something might have stuck.
The Persians might not attack Justinian so forcefully if he were not so ambitious. For one thing, the Persians would not be so tempted with the complete absence of the Byzantine army in the east, and for another, Khusrau Anushrivan feared that Justinian was re-uniting the entire old Roman empire for a massive invasion of Persia. Finally, in order to placate the Persians, he might just let them take Lazica (modern day Georgia) (instead of opposing them as IOTL) to compensate for their allowing him to take parts of the old Roman empire.
And even if Justinian focuses on Italy as some have suggested he do instead of OTL giving Belisarius the minimum amount of support, that's a fair amount of work for the Empire to bite off - especially since Justinian has multiple frontiers to juggle.
I find it amusing that all of your arguments directly contradict one another, stemming from Procopius' tendency to attack everything Justinian ever did without actually applying any logic. Justinian either exhausted the Empire by raising troops he didn't have,
or he stripped the East of troops,
or he put in minimal effort. He couldn't have done all three. As usual, the correct answer belongs to Elfwine, but it's a mistake to think that Belisarius in Italy was operating with an unusually small army- a generation later, the
Strategikon suggests that between five and ten thousand was a perfectly respectably sized army for warfare in the West.
Maybe they could restrict their more far flung conquests to strategic cities?
In Iberia for instance they just need a few forts and the Balaerics from which they can use their navy to try and control trade in the western med, will give them the advantage of a lot of income from the area without having to spend quite so much on administration and defence.
This is of course more or less what happened under Justinian's successors.
For one, you just need to forget about Hispania. Trying to take and hold it just wastes time and money. It's too far away from the empire's center of gravity to reliably supply, especially when it's surrounded by a semi-hostile power and has zero defensive hinterland. It's just one long line of coast.
One of the things you could do to hold Italy long term however, is to keep Justinian from shuffling about who's in charge of conquering the damn thing every week. Keep Belisarius in charge somehow (I dunno how you'd make Theodora less suspicious) and you have a much better chance of Italy remaining intact as a useful addition to the empire--not to mention a shorter war that wastes less manpower!
Africa's easy to hold, after all they did so perfectly well in OTL. You have Moors to worry about, sure, but nothing extraordinarily bad until...well, who even knows if the Muslims will even show up, or if Persia and Rome will have had it out as in OTL, or if they'll keep to border wars.
Well possibly, but Belisarius was recalled because he openly disobeyed a direct command by the Emperor to negotiate with the Ostrogoths in 540, instead seizing Ravenna and prolonging the war by another decade. Have Belisarius and his allies be a little less precocious in 540 and you have a restored Roman Italy that enjoys the benefit of a friendly buffer-state north of the Po.
Islam certainly won't emerge ITTL, but there could well be serious Arab expansion.
As much as it hurts to say this, kill Khosrau I and replace him with an incompetent. He was a good enough Shah that no matter what the eastern frontier is going to be a problem with as a competent Shah as that in charge, with a lesser enemy they would probably be much better off in the east.
This too. There seems to be a tendency in this thread to reduce the Iranians to passive figures in the sixth century, who behaved only to react to Roman behaviour, rather than as imperial actors in their own right. Khusrau is very likely to adopt an aggressive stance towards the old enemy, to unite Iran behind his far-reaching reforms of the whole state apparatus which were done largely to heal the wounds of Hephthalite invasion in the late fifth century and the temporary adoption of Mazdakism as a religion in the early sixth.