Indeed the byzantines called themselves romaioi, the romans and the arab term rumi: infidel is derived from it. You were aslo right to mention Justinian who reclaimed Italy for a few decades. But there are also differences. The first is the prevalence of greek instead of latin. While the old roman empire was (not counting local languages) bilingual with latin and greek, the new roman empire had lost almost all of his latin speaking territories. As a consequence the emperor Heraclius (early 7th century) got rid of the latin language, wich was no longer usefull. To your mention of Julius Caesar and Augustus, they were indeed seen as previous rulers but due to the importance of the christian religion, Constantine as the first christian emperor and founder of the capital was the man of choice, even if they still had some roman imperial jewels of the 1st century in the palace treasures. In addition, such texts as the gallic wars were in latin and thus were certainly not well known. Indeed with the hellenocentrification of the empire, a feeling of distinction from the western "latins" appeared, ending with the great schism. For the idea on the reconquest it's really difficult to judge because the roman empire was so much bigger that all of the area of action of the byzantine army was included in it. I can only think of one clear exception: the great counter offensive of Heraclius, striking at the heart of Sassanid empire to force them to leave roman soil. But this is not a very good example as they had not faced the islamo-arab conquest yet. In fact it's that long war against the Sassanians that exhausted both of them, clearing the path for the newly united arabs. You can also count the intervention in Egypt in the 12th century helping the crusaders as it was lost for 500 years but it went so wrong logistically that it ended in nothing but a loss of food and lives (they were not coordinated well enough and missed the opportunity to unite as one army).