Short answers, pulled from the story.
Modern scholars believe the physical institution did not exist until Ptolemy II Philadelphus took power in 283 BC. Plans for a universal library may have been proposed by Demetrius of Phalerum, an exiled Athenian statesman living in the city.
Eratosthenes of Cyrene served as head librarian from 280 BC to 194 BC. He calculated the circumference of the earth using mathematics applied to geography and produced a map of the known world incorporating data from Alexander's campaigns.
Julius Caesar besieged Alexandria in 48 BC during his civil war when his soldiers set fire to Egyptian ships docked in the port. The flames spread to nearby buildings including storehouses containing grain and books, with Seneca the Younger stating the fire destroyed 40,000 scrolls.
Mention of both the Great Library and the Mouseion disappeared after the mid-third century AD. Aurelian fought Zenobia's forces in 272 AD destroying the Broucheion quarter where the main library stood, and Diocletian's siege of Alexandria in 297 further damaged whatever remained.
The Serapeum temple housed a satellite collection called the daughter library which likely contained the largest number of books in Alexandria by the late fourth century AD. This library is attested until the Serapeum's final destruction along with its books in 391/392 AD.