Short answers, pulled from the story.
Count Franz von Walsegg commissioned Mozart's Requiem anonymously through intermediaries. Walsegg, an amateur musician who routinely purchased compositions and claimed them as his own, wanted a memorial mass for his wife Anna, who died on the 14th of February 1791 at the age of 20, so he could present the work as his own composition at her commemoration service on the 14th of February 1792.
No. When Mozart died on the 5th of December 1791, only the first movement, the Introitus, was fully complete. The Kyrie, Sequence, and Offertorium existed as detailed sketches, and the Lacrymosa breaks off after the first eight bars in Mozart's hand. The remaining movements were written entirely by other composers, primarily Franz Xaver Sussmayr.
Joseph von Eybler was the first composer asked to complete the Requiem; he worked on movements from the Kyrie through the Lacrymosa before returning the manuscript to Constanze Mozart. Franz Xaver Sussmayr then took over, completing the Lacrymosa, orchestrating the earlier sketches, and composing the Sanctus, Benedictus, Agnus Dei, and Communio. Sussmayr's version remains the standard performing edition.
During the 1958 World's Fair in Brussels, someone tore off the bottom right-hand corner of the second-to-last page of the autograph manuscript, folio 99r/45r. The stolen fragment contained Mozart's instruction "Quam olim d: C:", directing that the Quam olim fugue be repeated da capo. The perpetrator was never identified and the fragment has not been recovered.
At least 19 conjectural completions of Mozart's Requiem have been made, eleven of which date from after 2005. These alternatives were developed by composers and musicologists dissatisfied with Sussmayr's original completion, which has nonetheless remained the most widely performed version.
Handel was a primary influence: the Introitus theme is modelled on the opening chorus of Handel's 1737 Funeral Anthem for Queen Caroline, HWV 264, and the Kyrie fugue is based on the chorus "And with His stripes we are healed" from Handel's Messiah. Michael Haydn's Requiem in C minor also influenced the work; the theme of Mozart's Quam olim Abrahae fugue is a direct quote of the fugue theme from Haydn's Offertorium, a mass Mozart had performed at its premiere in January 1772.