Short answers, pulled from the story.
Pope Sixtus IV signed the papal bull Ad decorem militantis ecclesiae on the 15th of June 1475 to formally establish what is now known as the Vatican Library. The institution existed earlier through collections gathered by previous popes, but this document created it as an official entity.
The library began at the Lateran Palace until the end of the thirteenth century and later moved during the Avignon period when seven successive popes resided there between the death of Boniface VIII and the 1370s. Around 1587 Pope Sixtus V commissioned architect Domenico Fontana to construct a new three-story wing that now bisects Bramante's Cortile del Belvedere.
Pope Leo XIII formally reopened the library to scholars in 1883 following gradual lifting of barriers throughout the seventeenth century. Napoleon Bonaparte had seized and removed contents to Paris in 1809, but authorities returned the materials in 1817 after his defeat and abdication.
The oldest documents date back to the first century of recorded history with Codex Vaticanus Graecus 1209 standing as one of the oldest known nearly complete manuscripts of the Bible. The Secret History of Procopius was discovered here and published in 1623 while the largest manuscript measures 540x420mm and the smallest is a circle of 45mm diameter.
The library currently employs some 80 staff members working across five departments including manuscripts and archival collections, printed books and drawings, acquisitions and cataloguing, coin collections and museums, and restoration and photography. Father Franz Ehrle implemented the first modern catalogue system between 1927 and 1939 using the Library of Congress card catalogue method.