Short answers, pulled from the story.
The Ajanta Caves were built in two phases. The first phase ran from roughly the 2nd century BCE to the 1st century CE, producing caves 9, 10, 12, 13, and 15A. The second, far more prolific phase took place largely between 460 and 480 CE, according to the chronology developed by scholar Walter Spink.
British officer John Smith of the 28th Cavalry is credited with bringing the caves to Western attention on the 28th of April 1819, when a local shepherd boy guided him to the entrance of cave 10 during a tiger-hunting expedition. The caves were already well known to local people before Smith's visit.
The Ajanta Caves are Buddhist monuments, encompassing both the early Hinayana tradition and the later Mahayana tradition. The earlier phase of caves emphasised the stupa over figurative sculpture, while the second phase, linked to Mahayana Buddhism, featured large Buddha statues and elaborate painted Jataka tales.
The second phase of construction was primarily funded by associates of Hindu Emperor Harishena of the Vakataka dynasty, who reigned from 460 to 477 CE. Key patrons included his prime minister Varahadeva, who sponsored caves 16 and 17, and the sub-king Upendragupta, who sponsored at least five caves. Harishena himself commissioned cave 1.
Caves 1, 2, 16, and 17 form the largest surviving corpus of ancient Indian wall paintings. The murals are painted in dry fresco on plaster surfaces rather than wet plaster, and their compositions spread scenes in all directions from a central figure rather than in horizontal bands. Scholar James Harle described the best-preserved caves as representing the great glories not only of Gupta but of all Indian art.
Walter Spink argues that patronage collapsed around 480 CE following the death of Emperor Harishena in 477 CE and the subsequent revolt by the Asmaka kings, which ended the Vakataka dynasty. After 480 CE, according to Spink, no new images were created at the site, though a Rashtrakuta inscription outside cave 26 suggests the caves remained in use into the 7th or early 8th century.