Short answers, pulled from the story.
The earliest probable date for their meeting stands as December 22 and 23, 1783. This encounter occurred during a performance sponsored by the Vienna Tonkünstler-Societät. The two composers likely did not meet until after Mozart permanently relocated to Vienna in 1781.
Jens Peter Larsen suggests quartet playing formed the central element of contact between Haydn and Mozart. They participated together taking viola parts in Mozart's string quintets numbered K. 515, 516, and 593. Michael Kelly recorded an event where Haydn played first violin and Mozart took viola duties at a party hosted by Storace.
Mozart dedicated six string quartets numbered K. 387, 421, 428, 458, 464, and 465 to Haydn during their friendship years. He called Haydn his best friend and asked him to receive the six quartets kindly as his musical sons. These compositions appeared in print in 1785 after being written over several months.
Mozart joined the Zur Wohltätigkeit lodge on the 14th of December 1784 while Haydn applied to True Concord on the 29th of December 1784. Otto Jahn reported that Haydn and Mozart used informal du forms during conversation which indicated deep personal friendship. No evidence exists showing Haydn ever attended another meeting after his initial admittance ceremony before he was dropped from the lodge rolls in 1787.
On the 15th of December 1790, Haydn left with Salomon while Mozart never left his friend that day. They dined together before parting when Mozart said they were probably saying our last farewell in this life. Tears welled from both men's eyes as Haydn applied those words to himself.