Short answers, pulled from the story.
Idomeneo premiered on the 29th of January 1781 at the Cuvilliés Theatre in Munich, under the musical direction of Christian Cannabich. The opera had been commissioned by Karl Theodor, Elector of Bavaria, for a court carnival, and it received three performances in Munich.
The libretto was written by Giambattista Varesco, the court chaplain in Munich, who adapted a French text by Antoine Danchet. That text was based on a 1705 play by Crébillon père, which André Campra had already set to music as Idoménée in 1712.
Strauss replaced roughly one third of Mozart's score with his own music and used a German libretto by Lothar Wallerstein. He introduced the "Fall of Troy" motif from his 1928 opera Die ägyptische Helena and changed the character of Elettra into a priestess named Ismene.
The British premiere of Idomeneo was given on the 12th of March 1934 by the Glasgow Grand Opera Society under conductor Erik Chisholm. The first American performance followed in the summer of 1947, produced by Boris Goldovsky at the Berkshire Music Festival at Tanglewood.
K. 367 is a ballet divertissement Mozart composed for the Munich premiere of Idomeneo, lasting around fifteen minutes. It includes a Chaconne, a Pas seul in D major, a Passepied, a Gavotte in G, and an unfinished Passacaille in E-flat major. Jean-Pierre Le Grand served as the main choreographer for the original production.
Idomeneo is considered Mozart's first mature opera and the first in which he ended the work in the same key as its overture, a practice he maintained for every subsequent opera he wrote. The score displays his mastery of orchestral color, accompanied recitatives, and melodic line.