Anna's desire to be a dancer was sparked at the age of 8 when her mother took her to see a performance of The Sleeping Beauty at the Imperial Mariinsky Theatre. She auditioned for entrance into the Imperial Ballet School, now known as the Vaganova Ballet Academy, the same year but was turned down due to her age and physical appearance. Okay there’s not necessarily grammar errors here, but the wording in the last two sentences was rather awkward and I sort of couldn’t understand what you were trying to say until I reread it. I would omit some of the information and reword it to read something like: “Anna’s desire to be a dancer was sparked at the age of eight when she was taken by her mother to see a performance of The Sleeping Beauty. The same year, she auditioned for entrance into the Imperial Ballet School, but was declined due to her age and physical appearance.”

She auditioned a second time when she turned ten and was finally accepted into the school. She was at a physical disadvantage to the other students as a result of her thin ankles, long legs, and prominent arches. At the time, the current ideal body for a dancer was "small and compact.” I would omit “current” since it is a word which is meant to refer to the present. Also it’s redundant to “at the time” and therefore unnecessary. Taunts from her classmates did not deter her from pursuing a career as a ballerina. I would join “as a ballerina” and “they only fueled” together with a semicolon instead of a period, since the two sentences appear a bit stunted when separated. They only fueled her to practice extra hours to improve her technique. The use of “to” twice and in such close succession sounds a little funny. For a better flow, you could change it to something like “they only fueled her to improve her technique by putting in extra hours of practice.”

In 1899, Anna graduated from the academy and entered the Imperial Ballet as a coryphée, the rank above the corps de ballet. Her debut performance, Les Dryades prétendues, was praised by many critics. Pavlova danced in a style similar to that of the dancers from the Romantic era of ballet. She possessed a natural talent and passion for ballet, which often made up for any technical errors she made. One time in class, Anna was reported to have attempted to imitate there’s that “to” again— maybe “to have attempted and imitation of” the 32 fouettés en tournant, made famous by Pierina Legnani, the Prima Ballerina Assoluta of the Imperial Ballet. In response, her teacher Pavel Gerdt is quoted to have said, "... leave acrobatics to others. It is positively more than I can bear to see the pressure such steps put on your delicate muscles and the severe arch of your foot. I beg you to never again try to imitate those who are physically stronger than you. You must realize that your daintiness and fragility are your greatest assets. You should always do the kind of dancing which brings out your own rare qualities instead of trying to win praise by mere acrobatic tricks."

Anna became a favorite of Marius Petipa, the ballet master at the Imperial Ballet. He taught her various roles, including that of the title character in Giselle. Anna was promoted to Prima Ballerina following an outstanding performance of Giselle in 1906. It kind of sounds odd to have “Giselle” repeated twice so closely, so maybe you could join the two sentences. Anna performed with the Imperial Ballet until 1914, at the start of World War I. At that point, she created her own company and continued to tour until her death.

Anna is most noted for her role as the swan in The Dying Swan, a solo created for her by Mikhail Fokin in 1905. She performed the role at least 4,000 times during her career. Anna Pavlova was touring in the Netherlands when she contracted pneumonia. Doctors told her that she needed an operation to live, and that she would never dance again afterward. Anna refused the operation, proclaiming she would rather be dead than never dance again. She passed away three weeks later. Her reported last words were, "Get my swan costume ready." Then, "Play that last measure softly."