I can't speak much for the first question but as for the second, they actually considered themselves to be the successors of Rome, since the Byzantine Empire was in fact the remnants of the Eastern Roman Empire. So funnily enough they did view themselves as remnants of an ancient civilisation (there's a fair bit of debate as to whether or not they really were), but not the ones I think you were thinking of.

The name 'Byzantine Empire' actually came into use decades after its collapse by a historian, and before then they were mostly referred to by their contenporaries as 'the Greek Empire'. The name Byzantine was supposed to mimmick this sentiment. This was meant to be used as an insult and was used mostly to disregard their claims to be the successors to Rome (which Charlemagne and then Holy Roman Empire was already doing along with the Papacy).

If you don't know much about the history behind the Byzantines you might be interested to know that Emperor Justinian actually managed to retake Rome, Italy, and much of Northern African provinces.