Short answers, pulled from the story.
The King's Men was an English acting company active in the 17th century, formerly known as the Lord Chamberlain's Men during the reign of Queen Elizabeth I. King James I became their patron in 1603, granting them a royal patent and lending his name to the company. William Shakespeare belonged to the company for most of his career.
The King's Men received their royal patent on the 19th of May 1603, shortly after James I became king. The patent listed nine principal players by name, beginning with Lawrence Fletcher and including William Shakespeare and Richard Burbage, all of whom became Grooms of the Chamber as a result.
The 1603 patent named Lawrence Fletcher, William Shakespeare, Richard Burbage, Augustine Phillips, John Heminges, Henry Condell, William Sly, Robert Armin, and Richard Cowley. All nine became Grooms of the Chamber and were each given four and a half yards of red cloth for King James I's coronation procession in 1604.
The Blackfriars gave the King's Men an enclosed indoor venue allowing year-round performances, unlike the open-air Globe. Ticket prices at the Blackfriars ranged from sixpence to two shillings and sixpence, five to six times higher than at the Globe. The company acquired five of the seven Blackfriars shares in August 1608.
Squibs discharged during a performance of Henry VIII on the 29th of June 1613 set fire to the Globe Theatre's thatch roof, destroying the building. The King's Men rebuilt the Globe by the following spring at a cost of £1,400, replacing the thatch with tile. Because they also held the Blackfriars Theatre, they did not lose their playscripts or costumes in the fire.
Parliament ordered all London theatres to close on the 2nd of September 1642, at the start of the English Civil War. Two separate attempts to revive the company in 1648 both failed, the second still producing litigation as late as 1661. When the theatres reopened at the Restoration in 1660, a new King's Company was formed with little direct continuity from the original.