Short answers, pulled from the story.
After the fall of Constantinople in 1453, the most significant claimants in the East were the Ottoman Empire and the Russian Empire, both of which claimed succession of the Byzantine Empire. In the West, the Holy Roman Empire, which had operated since 800, continued its claim until its dissolution in 1806.
Mehmed II took the title Kayser-i Rum, meaning Caesar of the Romans, because Byzantine imperial ideology held that control of Constantinople was the key legitimizing factor for an emperor. The contemporary scholar George of Trebizond supported this claim, and Gennadius Scholarius, whom Mehmed installed as Ecumenical Patriarch in 1454, formally endorsed it.
The Third Rome concept holds that Moscow became the rightful successor of the Roman Empire after Rome and Constantinople both fell, with the First Rome being Rome in Italy and the Second Rome being Constantinople. In 1510, Russian monk Philotheus of Pskov articulated this in a letter to Grand Duke Vasili III, declaring that two Romes had fallen, the third stood, and there would be no fourth. The claim was reinforced when Ivan III married Sophia Palaiologina, a niece of the last Byzantine Emperor, in 1472, and when Moscow was granted its own patriarchate in 1589.
The term Byzantine Empire only came into general use outside France around the mid-19th century, after Barthold Georg Niebuhr and his continuators published the 50-volume Corpus Scriptorum Historiae Byzantinae. The word Byzantium itself had lain largely dormant since Constantine renamed the city Constantinople in 330, until the German scholar Hieronymus Wolf reused it in 1557 in a collection of historical sources about the Eastern Empire.
In the Islamic world the Ottoman claim was widely recognized. Documents from the Mughal Empire titled the Ottoman sultan as Qaiser-i-Rum, Caesar of Rome. The Persian chronicler Firishta referred to the Ottoman sultan as Sultan-i-Rum and Khunkar-i-Rum, Lord of Rome. Chinese usage during the Ming dynasty called the Ottomans Lumi, derived from Rumi, meaning Roman.
Mehmed II's Italian campaign began in 1480 with the invasion of Otranto, aiming to conquer Rome and reunite the Empire for the first time in nearly eight centuries. His sudden death on the 3rd of May 1481 cut the effort short. None of his successors renewed that attempt.