Short answers, pulled from the story.
Jean-Auguste-Dominique Ingres (1780-1867) was a French Neoclassical painter whose expressive distortions of form made him an important precursor of modern art, influencing Henri Matisse, Pablo Picasso, and other modernists. Although he aspired to be a history painter, his portraits are now recognized as his greatest legacy.
The Prix de Rome was the highest prize of the French Academy, entitling the winner to four years of residence at the French Academy in Rome. Ingres won the top prize in 1801 with his painting The Ambassadors of Agamemnon in the tent of Achilles, though his residence was postponed until 1806 due to shortage of state funds.
"Violon d'Ingres" is a French expression meaning a second skill beyond the one for which a person is mainly known. It comes from the painter Jean-Auguste-Dominique Ingres, who was a passionate amateur violinist throughout his life, having played second violin in the Orchestre du Capitole de Toulouse as a teenager and later performing Beethoven string quartets with Niccolo Paganini.
At the 1905 Salon d'automne retrospective in Paris, both Picasso and Matisse saw Ingres's work and acknowledged a direct debt. The Turkish Bath influenced the poses and composition of Picasso's Les Demoiselles d'Avignon in 1907, and Ingres's studies of women dancing in a circle inspired Matisse's La Danse in 1909. Picasso's Portrait of Gertrude Stein (1906) also echoed the pose and colouring of Ingres's Portrait of Monsieur Bertin.
Ingres and Delacroix became the most prominent representatives of the two competing schools of French painting in the mid-19th century: neoclassicism and romanticism. Their public rivalry intensified at the Salons of 1824 and 1827. According to Ingres's student Paul Chenavard, the two men later accidentally met on the steps of the French Institute, shook hands amicably, and parted.
The Turkish Bath is a circular painting Ingres signed and dated in 1862, depicting numerous female nudes. It was originally completed in a square format in 1852 and sold to Prince Napoleon in 1859, but returned to the artist, reportedly because Princess Clothilde was shocked by the abundant nudity. After reworking, it was offered to the Louvre in 1907 but rejected, and finally accepted by the Louvre in 1911.