Short answers, pulled from the story.
The Tonkünstler-Societät was a benevolent society for musicians founded in Vienna in 1771 by Florian Gassmann. Its stated purpose was to support retired musicians and their families, specifically their widows and orphans. It funded this mission through a series of large-scale benefit concerts held twice at Easter and twice before Christmas each year.
Florian Gassmann founded the Tonkünstler-Societät in 1771. The Society received early backing from Empress Maria Theresa, who authorized its founding on the 23rd of February 1771 and made an initial contribution of 500 ducats (about 2000 florins) to its fund.
Haydn was the composer most closely associated with the Tonkünstler-Societät. He composed his oratorio Il ritorno di Tobia for it and led its premiere in 1775. After a period of friction over membership terms, he became an honorary member in 1797. He personally directed the Society's first performances of The Creation in December 1799 and The Seasons in December 1801, and the Society renamed itself after him in 1862.
Mozart's application stalled because the Society required him to produce his birth certificate, and having been born in Salzburg made that difficult to arrange. He promised twice to provide the document but never did. He remained associated with the Society as a performer and commissioned composer but was never enrolled as a member when he died in 1791.
Mozart's Clarinet Quintet received its first known public performance at a Tonkünstler-Societät concert on the 22nd of December 1789 at the Burgtheater in Vienna. The clarinettist was Mozart's friend Anton Stadler.
The Tonkünstler-Societät was abolished on the 9th of March 1939 by the National Socialist government of Germany, which had annexed Austria the previous year in the Anschluss. The Society had operated continuously for nearly 170 years before it was dissolved by government decree.