Short answers, pulled from the story.
The name derives from the Temple of Jupiter Optimus Maximus built on the hill's summit, known as the Capitolium. Ancient sources connect the name to the Latin word caput, meaning head or summit, citing a legend that a man's head was found when foundation trenches were dug for the temple. The hill was earlier called Mons Saturnius, dedicated to the god Saturn.
Tarpeia was a Roman maiden who let the Sabines into the citadel on Capitoline Hill, and was punished by being thrown from a steep cliff overlooking the Roman Forum. That cliff was named the Tarpeian Rock in her memory and became a regular execution site. Tarpeia is described in ancient sources as the first person to be punished in this way at that location.
Michelangelo Buonarroti designed the Piazza del Campidoglio and the surrounding palazzi between 1536 and 1546. He was commissioned by Pope Paul III, who wanted to impress Emperor Charles V. Most of the design was completed in the 17th century after Michelangelo's death, and the paving design was not finished until Benito Mussolini ordered it completed in 1940.
The most important was the Temple of Jupiter Optimus Maximus Capitolinus, built in 509 BC and described as nearly as large as the Parthenon. Also built on the hill were the temple of Juno Moneta and the temple of Virtus. The Temple of Saturn stood at the foot of the hill at the western end of the Forum Romanum.
The Tabularium is a building constructed in the first century BC to hold the official records of the Roman state. It sits underground beneath the hilltop piazza and looks out from its rear onto the Roman Forum. Its main interior feature, besides the structure itself, is the Temple of Veiovis.
Capitol Hill in Washington, D.C. is widely assumed to be named after the Capitoline Hill in Rome. The English word capitol itself derives from the Latin word Capitolium, which was the original name for the Temple of Jupiter Optimus Maximus and later came to refer to the entire hill.