Short answers, pulled from the story.
Roman Egypt as a province began in 30 BC, when Octavian defeated Mark Antony and Cleopatra and seized the Ptolemaic Kingdom. It ended in AD 642, when the Muslim conquest transferred Egypt to the Rashidun Caliphate.
A law promulgated by Augustus required that the governor of Egypt hold equestrian rank rather than the senatorial rank used in other provinces. This prevented any governor from building a rival power base, and it meant the legion commanders in Egypt were equestrians too, an arrangement unique in the Roman Empire. Senators were not even permitted to enter Egypt without the emperor's personal approval.
Roman citizens and Alexandrian citizens held the highest status and were exempt from the poll tax. Metropolitan Greeks paid a reduced poll tax rate. Native Egyptians occupied the lowest class and paid the highest tax rates, roughly three to five times more per unit of land than elites or Alexandrians. Native Egyptians were also barred from serving in the Roman legions, though they could join auxiliary units and receive citizenship on discharge.
Alexandria was the capital of the province and the second largest city in the Roman Empire. It was the seat of the prefect of Aegyptus, the base of the provincial fleet, and the main port for grain shipments to Rome. The city was legally distinguished from Egypt proper, which is why the governor's full title named both Alexandria and Egypt separately.
Christianity probably arrived among Hellenized Alexandrian Jews from Palestine. By around 200 AD, Alexandria was a major Christian center, home to theologians including Clement of Alexandria and Origen. The Arian controversy of the 4th century triggered years of riots, and the Miaphysite dispute following the Council of Chalcedon in 451 caused the Church of Alexandria to split from Rome and Constantinople, producing the Coptic Orthodox Church of Alexandria.
The Justinianic Plague was the first pandemic of bubonic plague to reach the Mediterranean world. It emerged at Pelusium in Roman Egypt in 541, making Egypt the point of entry for the First Plague Pandemic into the broader Roman Empire.