Were only those born in the city considered citizens, or would a Sicilian farmer, and Macedonian merchant consider themselves Romans? Would their answers be different during the republican and imperial periods?
If more than just Roman’s could be “Roman” was this a privilege afforded to everyone, only to people from certain provinces (like statehood), or only certain ethnic groups?
It very much depends on the period. During the Republic, only the children of Roman citizens were citizens and that generally meant being from Rome or its immediate surrounds. The rest of Italy was made up of Latin and Italian communities, with their own treaties and rights at Rome. Following the Social War (91-89 BCE) the rest of the Italians, including the Greek cities in the south, got Roman citizenship. It was gradually spread through out the peninsula (Caesar enfranchised some communities from the north that were considered "Gauls"). The Emperor Claudius extended citizenship to some communities in modern Germany and emperors continued granting citizenship as a favour to individuals or communities until Caracalla made all freeborn inhabitants of the empire citizens in 212 CE. The citizenship brought various privileges- the right to marry a Roman and have Roman children, the right to bid on some public contracts, freedom from taxation, the right to vote, run for office, access the corn dole (once this was established by Gaius Gracchus) and also protection from flogging and crucifixion. If you're interested in more about how Latins and Italians could get citizenship pre 91, I can tell you about that :)