Short answers, pulled from the story.
The Bibliothèque nationale de France traces its origins to the royal library founded at the Louvre Palace by Charles V in 1368. Charles received a collection of manuscripts from his predecessor, John II, and appointed Claude Mallet as the first librarian of record.
At the death of Charles VI, the English regent of France, the Duke of Bedford, unilaterally purchased the entire collection and transferred it to England in 1424. It was apparently dispersed following the Duke's death in 1435.
The BnF holds roughly 14 million books across its four Parisian sites, along with manuscripts, photographs, maps, coins, audio recordings, video documents, and theatrical sets. Its digital library Gallica contains more than ten million freely available items.
The library opened to the public in 1692, under the administration of Abbott Camille le Tellier de Louvois. Before that it had operated as a royal collection accessible only to the Crown and its agents.
Gallica is the BnF's free online digital library, established in October 1997. It reached ten million documents on the 30th of March 2023 and includes books, manuscripts, newspapers, images, maps, audio recordings, and video materials.
The building acquired the sarcastic nickname TGB, or Très Grande Bibliothèque, because its construction ran into severe cost overruns and technical difficulties. The name mocked the acronym of France's celebrated TGV high-speed rail system. The library was inaugurated on the 15th of December 1996.