Short answers, pulled from the story.
A sketch in music is an informal document a composer prepares to help shape a piece, ranging from a few notes to a nearly finished draft. Whether it survives depends largely on the composer's own habits, from Johannes Brahms discarding his sketches to Beethoven keeping his in great numbers.
Mozart kept a great many sketches during his lifetime, but after his death some were given away to friends as keepsakes and were subsequently lost.
Composers sketch partly because human memory is unreliable, and partly to plan the architecture of a piece in advance. Eighteenth century theorists H. C. Koch and J. G. Sulzer advised composers to lay out how a movement's themes would fit together, often within the pattern known as sonata form, before finishing the work.
Antony Hopkins observed that an early sketch for the main theme of Beethoven's Second Symphony 'bears more resemblance to a bugle-call than a symphony.' He added that the sketch still seemed to be 'an essential part of his creative procedure,' and traced, across pages 38-39 of his discussion, how two further sketches evolved toward the finished theme.
Surviving Bach sketches include a chorale prelude fragment, O Traurigkeit, o herzeleid, BWV Anh. 200, found in the Orgelbüchlein manuscript, and a six bar cantata outline, BWV Anh. 2, for Trinity XIX. Also among them is BWV 149/1a, an opening movement that breaks off after the word 'Man,' and a nine bar fragment, BWV 1059, of a harpsichord concerto.
Franz Schubert left at least four independent sets of sketches for a symphony in D major, catalogued as D. 2B, formerly D 997, D 615, D 708A, and D 936A, the last known as Symphony No. 10. Apart from these, he completed one D major symphony, the Third Symphony, D 200.