Short answers, pulled from the story.
Russian ballet reached Paris in a significant way by the early 1900s. In 1903, Ivan Clustine, a Russian dancer and choreographer who had started his career at the Bolshoi Theatre, was appointed Maître de ballet at the Paris Opera. Sergey Diaghilev then founded the Ballets Russes in Paris in 1909.
Sergey Diaghilev, who lived from 1872 to 1929, founded the Ballets Russes in 1909 and headquartered the company in Paris. He was involved in every aspect of production, including direction, lighting, scenery, and performance.
The Vaganova method is the most widely used pedagogical approach in Russian ballet. It is named after the ballerina and teacher Agrippina Vaganova. The Vaganova Academy of Russian Ballet was originally founded in 1738 as the Imperial Ballet School.
Peter the Great opened Russian society to Western influence and built St. Petersburg to rival Western courts. Under his vision, classical ballet entered Russia not as entertainment but as a standard of physical comportment meant to cultivate a new kind of Russian person.
The rayok, sometimes called the "paradise gallery," was a seating section in Imperial Russian theaters that consisted of simple wooden benches at inexpensive prices. It allowed non-wealthy people to attend ballet performances for the first time in the early nineteenth century.
George Balanchine was a protégé of Sergey Diaghilev. Balanchine went on to found the New York City Ballet in 1948, tracing a direct lineage from Imperial Russian ballet through the Ballets Russes to the United States.