Short answers, pulled from the story.
Gregorio Allegri composed his setting of Psalm 51 during the reign of Pope Urban VIII likely in the 1630s. The work was created for exclusive use within the Sistine Chapel during Tenebrae services of Holy Week.
A letter dated the 14th of April 1770 describes how fourteen-year-old Wolfgang Amadeus Mozart heard the piece during a Wednesday service in Rome. He reportedly wrote it down entirely from memory later that same day.
Pietro Alfieri published an edition in 1840 that included ornamentation to preserve performance practices of the Sistine choir. The original ornamentations were Renaissance techniques closely guarded by the Vatican and virtually unknown outside its walls.
Ivor Atkins released his English translation in 1951 replacing the Latin text with Miles Coverdale's version from the Book of Common Prayer. A recording based upon this translation by the Choir of King's College Cambridge popularized the piece globally.
In 2015 the Sistine Chapel Choir released their first CD including the 1661 Sistine codex version recorded inside the chapel itself. This release marked a significant moment for preserving the authentic sound of the work.