Short answers, pulled from the story.
Caesar derives from the cognomen of Gaius Julius Caesar, the Roman dictator assassinated in 44 BC. After his death, his adopted son Octavius took the name, and it passed through every subsequent Roman emperor until it became synonymous with imperial rule.
The shift can be traced to AD 68, following the death of Nero and the fall of the Julio-Claudian dynasty. Galba took the throne as "Servius Galba Caesar Augustus" without any familial claim to the name, establishing it as a title rather than an inherited surname.
In the Byzantine Empire, Kaisar was a senior court dignity, originally used for the heir apparent. After Alexios I Komnenos created the sebastokrator in 1081 and Manuel I Komnenos created the title of despot between 1143 and 1180, caesar fell to third place in the court hierarchy, where it remained until the fall of Constantinople.
Following his conquest of Constantinople in 1453, Mehmed II assumed the title Kayser-i Rûm, meaning Caesar of the Romans, on the grounds that possession of the city made him the rightful successor to the Roman emperors. The Ecumenical Patriarch Gennadius II, enthroned by the Sultan in 1454, formally recognized this claim.
German Kaiser, Russian Tsar, Polish cesarz, Czech císař, and Hungarian császár all derive from Caesar. These words trace the spread of Roman imperial concepts across Europe and the Balkans.
Under the Treaty of Constantinople of 1533, the Austrians of the Holy Roman Empire agreed to relinquish the use of the Caesar title, which the Ottomans saw as their exclusive inheritance as successors of Rome. In practice, the Austrians continued using it until the Holy Roman Empire collapsed in 1806.