I'm trying to get a feel for how institutions changed during the rise of Islam, so let's say circa 600, but I'll take what I can get.
Short answer to your first question - the city of Rome got its grain provisions from Egypt through an annual tax, levied on the harvest, known as the annonae. From the 4th century on, Constantinople was also provisioned the same way. With the Islamic conquest of Egypt, Constantinople switched to Thrace and Bithynia for its grain supply.