Short answers, pulled from the story.
The citizens of the Eastern Roman Empire called themselves Romans. They viewed their state as the direct continuation of the ancient Roman state and never used the term Byzantine to describe themselves.
The Byzantine Empire existed from the 330 to 1453. This period spans over a thousand years of continuous identity as Romans before the fall of Constantinople to the Ottoman Turks.
The Plague of Justinian began in the year 541 and killed a large proportion of the empire's population. This outbreak reduced the population from 27 million at its peak in 540 to just 12 million by 800 and severely disrupted the economy.
The Arab conquests of the 7th century took the richest provinces of Egypt, Syria, and the Levant from the Byzantine Empire. These territories had provided the empire with the majority of its revenue and their loss deprived the state of as much as three-quarters of its income.
The Ottoman Turks led by Sultan Mehmed II captured Constantinople in the year 1453. This event marked the end of the Byzantine Empire and the death of the last emperor Constantine XI in battle.