Short answers, pulled from the story.
The title augustus was granted to Octavian by the Roman Senate in January 27 BC, the date most modern historians use as the start of the Roman Empire. Most Romans simply regarded the emperor as whoever held supreme power, with no single formal title defining the office.
Modern historians regard Augustus (Octavian) as the first Roman emperor. Most ancient Roman writers, including Pliny the Younger, Suetonius, and Appian, considered Julius Caesar the first emperor, while Plutarch, Tacitus, and Cassius Dio agreed with the modern view that Caesar was the last dictator of the Republic.
Legitimacy depended on control of the Roman army and recognition by the Senate, though in practice it was often determined retrospectively. An emperor's proclamation by troops, distribution of coins and statues, claimed blood or adoptive connections to past emperors, and propaganda were all used to assert legitimacy, by both recognized emperors and usurpers alike.
Romulus Augustulus was overthrown by Odoacer in 476, the date historians mark as the fall of the Western Roman Empire. Julius Nepos, who had been displaced by Romulus, continued to claim the title from Dalmatia until his murder in 480, and the Eastern court recognized his claim until his death.
The principal titles were imperator, caesar, augustus, princeps, and, in the Eastern Empire from 629 onward, basileus. The formula Imperator Caesar followed by the full name and Augustus became standard from the reign of Antoninus Pius. In the Eastern Empire the full title eventually became basileus and autokrator of the Romans.
Constantine XI Palaiologos was the last Eastern Roman emperor. He died in battle during the fall of Constantinople to the Ottoman Turks in 1453. The last remnants of the empire, Morea and Trebizond, fell in 1461.