Short answers, pulled from the story.
Ludwig van Beethoven arrived in Vienna in January 1787 and departed around March or April of that year. Records indicate he returned to Bonn because his mother fell ill with tuberculosis and died in July.
Written documentation about this initial visit is sparse enough that historians debate whether the composers ever met. Contemporary scholarship remains skeptical about accounts claiming they encountered each other, though The New Grove Dictionary suggests they likely met.
Beethoven wrote cadenzas WoO 58 to the first and third movements of Mozart's D minor piano concerto K. 466. He also created four sets of variations on themes from Mozart including Se vuol ballare from The Marriage of Figaro and Là ci darem la mano from Don Giovanni.
Ludwig van Beethoven was born in Bonn in 1770, fourteen years after Wolfgang Amadeus Mozart entered the world in Salzburg. This age gap meant Beethoven was sixteen when he made his first documented attempt to visit Vienna in 1787.
Charles Rosen sees Mozart's C minor Piano Concerto K. 491 as a model for Beethoven's 3rd Piano Concerto in the same key. Robert Marshall identifies Mozart's Piano Sonata No. 14 in C minor K. 457 as the model for Beethoven's Pathétique Sonata Op. 13.