Short answers, pulled from the story.
Kapellmeister comes from Kapelle, meaning chapel, and Meister, meaning master, so it translates literally to master of the chapel choir. It designates the leader of an ensemble of musicians, a meaning that has evolved from chapel choirs to today's smaller ensembles for television, radio, and theatre.
Johann Sebastian Bach served as Kapellmeister for Prince Leopold of Anhalt-Cöthen from 1717 to 1723. Joseph Haydn held the post for the Esterházy family for many years, and George Frideric Handel served as Kapellmeister for George, Elector of Hanover, who later became King George I of Great Britain.
No, Mozart never officially held the title of Kapellmeister. He received a paid position as Kammercompositeur, chamber composer, at the Austrian court in 1787, while Antonio Salieri held real musical authority; Mozart later applied for the Kapellmeister post at St. Stephen's Cathedral in April 1791 but died in December 1791 before the ailing incumbent, Leopold Hofmann, who did not die until 1793.
The number of Kapellmeister positions declined because many noble families lost economic power to the newly prosperous middle class and could no longer afford to maintain a Kapelle. In 1790, for example, Prince Anton Esterházy succeeded his father Nikolaus and dismissed almost all of his extensive musical establishment.
Today, Kapellmeister is less common than the German word Dirigent, meaning conductor, but when used it designates the director or chief conductor of an orchestra or choir. It also implies involvement in policy such as selecting repertoire and concert schedules, and in military settings it refers to a bandmaster.
The Kapellmeister Tradition describes conductors who embody knowledge of a work, great ability, and dedication to music, qualities Christian Thielemann says the term implies. Otto Klemperer, Clemens Krauss, and Erich Kleiber are cited as examples of conductors who paid their dues on their way to international fame.