Short answers, pulled from the story.
Whether Beethoven and Mozart met remains uncertain. Beethoven visited Vienna in January 1787 and stayed roughly six weeks, and the New Grove Dictionary says he likely met Mozart and may have had a few lessons from him. Some historians, including biographer Maynard Solomon, remain skeptical and suggest Mozart may have auditioned and rejected the young composer.
Beethoven was sixteen years old when he traveled to Vienna in January 1787, around fourteen years younger than Mozart. He departed in March or April of that year, returning to Bonn partly because his mother was gravely ill; she died of tuberculosis in July 1787.
Mozart's influence on Beethoven is documented across multiple works. Scholar Charles Rosen identifies Mozart's K. 491 Piano Concerto as a model for Beethoven's Third Piano Concerto, and Mozart's K. 452 Quintet as a model for Beethoven's Op. 16. Beethoven also wrote four sets of variations on Mozart themes and composed cadenzas for Mozart's D minor Piano Concerto, K. 466.
Beethoven described Mozart's playing as fine but choppy, using the German word zerhacktes, and said Mozart lacked ligato. He relayed this assessment to his student Carl Czerny, who passed it on to the biographer Otto Jahn.
Beethoven wrote four sets of variations on Mozart themes: one on 'Se vuol ballare' from The Marriage of Figaro (1792-93), one on 'La ci darem la mano' from Don Giovanni, and two on arias from The Magic Flute, including one completed in 1801. He also drew on Don Giovanni's opening aria by Leporello for the 22nd of the Diabelli Variations.
Beethoven, who settled permanently in Vienna in 1792 the year after Mozart died, followed a remarkably similar path. Both were mentored by Joseph Haydn, patronized by Countess Maria Wilhelmine Thun and Baron van Swieten, and both composed for soprano Josepha Duschek in Prague. Both also had works associated with the impresario Emanuel Schikaneder.