Short answers, pulled from the story.
Claude of France (the 13th of October 1499 - the 26th of July 1524) was the suo jure Duchess of Brittany and Queen of France as the wife of Francis I. She was the elder surviving daughter of King Louis XII and Anne, Duchess of Brittany.
Claude was heir presumptive to the Duchy of Brittany, and competing factions fought over whether she should marry the future Holy Roman Emperor Charles V, which would keep Brittany separate from France, or Francis, Duke of Valois, which would tie Brittany to the French crown. Her 1514 marriage to Francis ultimately secured Brittany's union with France.
Yes. After Francis became king in 1515, Anne Boleyn stayed as a member of Claude's household. Renée of France later described her as one of Claude's maids of honour. Anne Boleyn returned to England in late 1521 and eventually became Queen of England as the second wife of Henry VIII.
Claude and Francis I had seven children: Louise, Charlotte, Francis (Duke of Brittany), Henry (later Henry II of France), Madeleine, Charles, and Margaret. Several died young, and Henry II was the child who ultimately succeeded Francis I as king.
Claude died on the 26th of July 1524 at the Château de Blois at age twenty-four. The exact cause was disputed: sources variously attributed her death to childbirth, exhaustion from repeated pregnancies, bone tuberculosis, or syphilis contracted from her husband Francis I.
After Claude's death in 1524, the duchy passed to her eldest son Francis, who became Duke Francis III under his father's guardianship. When Francis died in 1536, Claude's second son Henry became Duke of Brittany and later ascended the French throne as Henry II.