Great question!

I am not as familiar with commoners' conventions (although I will see if I can find any good sources).

/u/bukoleon has a great concise response in their third paragraph.

As for the emperors, many often had attached epithets, both negative and positive: Julian "the Apostate" (4th century), Leo IV "the Khazar" (8th century), Basil II "the Bulgar Slayer" (10th/11th centuries), and a bunch had "the great." Or the (in)famous Constantine V, "the Dung-named" (don't try to take away people's religious images). This did help distinguish emperors. Generally speaking, most emperors pre-11th century did not use family names, such as Kommenos, Doukas, or Angelos.

Not a "source" per se, but a fun list of the emperors with their names exists on Wikipedia. https://en.wikipedia.org/wiki/List_of_Byzantine_emperors

It depends on a lot of factors. Surnames along with the whole complex Roman naming system disappeared in the early period and only started to reemerge in the 8th century. But these could be family names, patronyms, or toponyms. And due to Byzantine naming conventions (you'd typically name children after your parents), there were many common names like Constantine, John, Michael, Leo, etc. within the same family. So the Byzantines needed some way to distinguish between all these similarly named people.

For personal relations and commoners, I covered it briefly here. Within a local sphere, people would know you personally.

For higher-ups, it was much easier: the notables of the Empire were granted a plethora of ecclesiastical, civil, and courtly titles to distinguish themselves with. John the Orphanotrophos, the Caesar John Doukas, Leo the Deacon. So if you were a bureaucrat and you wanted to refer to Michael Cydones, who was a patrician, you'd might call him Michael the Patrician if he was a particularly notable patrician named Michael. Or if you had to be more specific, the patrician Michael Cydones.