Short answers, pulled from the story.
The Duchy of Milan was officially founded on the 11th of May 1395, when Gian Galeazzo Visconti received the title of Duke of Milan through a diploma signed in Prague by Wenceslaus of Bohemia. The nomination was ratified and celebrated in Milan on the 5th of September 1395.
Francesco Sforza, a mercenary captain who had previously married Bianca Maria, the illegitimate daughter of the last Visconti duke Filippo Maria, conquered Milan in 1450 after the short-lived Ambrosian Republic collapsed. His takeover restored the duchy and founded the House of Sforza.
King Louis XII of France claimed the duchy because his ancestor Louis of Orleans had married Valentina Visconti in 1389, with a marriage contract stating that the dukedom would pass to Valentina's descendants if the Visconti dynasty died out. Louis XII used this claim to invade in 1499.
Charles V gained control of the Duchy of Milan after Francesco II Sforza died without heirs in 1535. He installed his son Philip II with an imperial diploma signed in Brussels on the 11th of October 1540, and Philip's possession was formally recognized by France in the Treaty of Cateau-Cambrésis in 1559.
During the Franco-Spanish War from 1635 to 1659, Milan sent an average of 4,000 soldiers per year to the Spanish crown, contributing over 100,000 men in total. Many of these soldiers served in the Low Countries against the Dutch States Army.
Napoleon Bonaparte captured the duchy in 1796, and Austria ceded it through the Treaty of Campo Formio in 1797, leading to the formation of the Cisalpine Republic with Milan as its capital. After Napoleon's defeat, the Congress of Vienna on the 9th of June 1815 chose not to restore the duchy, folding its territory into the Kingdom of Lombardy-Venetia instead.