For example, would, say, a shepherd living in Gaul during the reign of Trajan or Hadrian have seen himself primarily as Gaulish, Roman, or something else? How about a farmer in Egypt?
Also, I've read that the Byzantines viewed themselves as ethnically Roman, but I get the impression that in earlier times the Greeks were seen as a separate culture from the Romans. When and how did this merger between Greek and Roman identity take place? Did anyone still view themselves as Greek but not Roman in the Byzantine Empire?
Did language play a role in identity? Did speakers of Coptic or Aramaic see themselves as different from the Latin- and Greek-speaking elite of the empire?
I can try examining this from the lense of art and architecture in the empire. Ethnic identity was something that varied within each region of the empire, and was something not too tied to actual ethnicity, but more towards culture. That being said, these individual cultures were far more prevalent in the eastern Hellenic provinces.
I know much more of the eastern culture, but much of the romanization of the west took place from military resettlement of Roman soldiers and overall lack of infrastructure prior to the Roman conquest. From this, Roman style buildings were much more common in the west than in the east. And a great example of the new Roman identity in the west would be that of the emperors Septimius Severus (Lepcis Magna in modern Libya) and Trajan (Spain), in which both were considered quite Roman. In contrast though there was the emperor Hadrian, who did much to strengthen Greek culture and art in the empire despite being a non-Greek.
In the eastern regions of the empire, much of the Greek culture from the Macedonian empire still remained. Obviously this persisted as is the case with the Byzantine empire. And you're quite right in saying the early Greeks thought of themselves separate to the Italian Romans. A professor I had once made the comparison in that Romans thought of Greeks and Anglos thought of the French, in that both were more sophisticated cultures.
Heres two artistic examples of Greek culture during the roman empire.
The Library of Celsus in Ephesus, Turkey, (2nd century AD) built by a wealthy Greek in which he is honored as both a Greek and Roman
The introduction of the beard into Roman portraiture by Hadrian, this was meant to invoke a sense of Greek heroism and strength and was common in Greek art but not in Roman
You mentioned Egypt as another area which I'd like to touch on as it is a very interesting case. Due to persisting Hellenism and the previous Egyptian culture, the two fused in a way. I can't really answer for poorer farmers, but the best example of this for the wealthy would be the Greco-Egyptian mummies during the imperial period.
Source: Fred S. Kleiner, A History of Roman Art
