Short answers, pulled from the story.
Neo-Latin, also called New Latin or Modern Latin, is the style of written Latin used in original literary, scholarly, and scientific works from the Italian Renaissance onward. It began in Italy in the fourteenth and fifteenth centuries and spread across northern Europe after about 1500. Productive use of Latin for most purposes ended in the early 1800s, though it persisted in specialised fields and official contexts well into the nineteenth century.
Petrarch, Salutati, Bruni, Ficino, and Pico della Mirandola were prominent early Neo-Latin writers in Italy. In northern Europe, key figures included Erasmus, Thomas More (whose Utopia appeared in 1516), Grotius, George Buchanan, and Elizabeth Jane Weston, the most well known female Neo-Latin poet, who published her Parthenica in 1608.
Neo-Latin was the primary language of international scientific publication during the sixteenth and seventeenth centuries. Major works including Copernicus's De Revolutionibus (1543), Galileo's Sidereus Nuncius (1610), Harvey's Exercitatio Anatomica (1628), and Newton's Principia Mathematica (1687) were all written in Neo-Latin. Carl Linnaeus used Neo-Latin as the basis for binomial nomenclature, the system still used to classify living organisms today.
Latin was the dominant language of international diplomatic correspondence through most of the seventeenth century. The Treaty of Vienna in 1738 and the Treaty of Belgrade in 1739 were among the last major international treaties written in Latin. After the War of the Austrian Succession (1740-48), international diplomacy shifted predominantly to French.
Neo-Latin had no single pronunciation. Each region adapted Latin sounds to match the dominant local language, producing distinct Western and Eastern dialect families. The Western family covered most Romance-speaking regions and the British Isles; the Eastern family covered Central Europe, Eastern Europe, and Scandinavia. Italian pronunciation was generally regarded as having higher status and acceptability among educated speakers.
Several forces combined: vernacular languages became better established and attracted larger national readerships, translation between vernaculars became more practical, and the utility of an auxiliary language diminished as French, German, and English became more widely known. Education also broadened to serve the middle and lower classes, for whom learning Latin offered limited practical benefit. French replaced Latin as the language of diplomacy in the early eighteenth century following the influence of Louis XIV's France.