Short answers, pulled from the story.
Twelve-tone technique is a compositional method Schoenberg announced in 1923 in which all twelve pitches of the octave are treated as equal, with no single note or tonality given the emphasis it occupies in traditional harmony. He described it as "a method of composing with twelve tones which are related only with one another". Berg, Webern, and Eisler adopted it, and later composers such as Boulez and Stockhausen extended its logic further.
Schoenberg taught first at the Malkin Conservatory in Boston after arriving on the 31st of October 1933, then moved to Los Angeles. He was appointed visiting professor at UCLA in 1935 and promoted to full professor in 1936 at a salary of $5,100 per year. He bought a home at 116 North Rockingham in Brentwood Park in 1937 and taught at both UCLA and USC, each of which later named facilities in his honor.
Schoenberg was warned while visiting France in 1933 that returning to Germany after the Nazi seizure of power would be dangerous. He formally returned to Judaism at a Paris synagogue before immigrating to the United States with his family, arriving on the 31st of October 1933. Nazi cultural policy had already denounced his music as degenerate art.
Schoenberg suffered from triskaidekaphobia, a fear of the number thirteen. According to his friend Katia Mann, he feared he would die during a year that was a multiple of thirteen. In 1950, on his seventy-sixth birthday, an astrologer pointed out that seven plus six equals thirteen, a calculation he had never considered before, which stunned and depressed him. He died on Friday, the 13th of July 1951, at approximately 11:45 pm.
Pierrot lunaire, Op. 21 (1912) is a song cycle Schoenberg composed for a touring actress who wanted a suitably unconventional setting of Albert Giraud's poems in Otto Erich Hartleben's expressionist German translation. Schoenberg devised the standard Pierrot ensemble of five players and introduced Sprechstimme, a voice technique realizing pitch only fleetingly. Harold C. Schonberg called it as seminal a work as Le Sacre du Printemps, Joyce's Ulysses, and Picasso's Les Demoiselles d'Avignon.
Schoenberg taught Alban Berg and Anton Webern, who became the core of the Second Viennese School alongside him. Later students included Hanns Eisler, Roberto Gerhard, Nikos Skalkottas, John Cage, and Lou Harrison. His pupils went on to teach at major institutions including Harvard, UCLA, USC, Oberlin, and the Juilliard School.