Short answers, pulled from the story.
The Lord Chamberlain's Men were an English playing company formed around 1594 under the patronage of Henry Carey, 1st Baron Hunsdon. They were the company for which William Shakespeare wrote during most of his career, and by 1603 they had become one of the two leading companies in London before becoming the King's Men under James I.
Shakespeare was a sharer in the Lord Chamberlain's Men, meaning he split profits and debts with seven other core members. He wrote the great bulk of the company's repertory and continued to act with them at least until 1603. The company held exclusive rights to perform his plays.
The Globe Theatre was built in 1599 from beams salvaged from an earlier playhouse called The Theatre in Shoreditch. On the 29th of December 1598, the Burbage brothers and a crew of workers dismantled The Theatre and carried the timbers south to Southwark. The Globe was destroyed by fire on the 29th of June 1613, rebuilt by June 1614, and finally closed in 1642.
Supporters of the Earl of Essex paid the company 40 shillings above their usual fee to perform Shakespeare's Richard II on the 7th of February 1601, the day before Essex's uprising, hoping the depiction of a deposed king would sway public opinion. Augustine Phillips testified that the company accepted only because of the extra payment. The authorities accepted this explanation and the company went unpunished.
Richard Burbage was the lead actor of the Lord Chamberlain's Men. He played Hamlet, Othello, King Lear, Macbeth, and many other leading roles. Relatively unknown in 1594, he became one of the most famous actors of the Renaissance, with a fame and wealth exceeded only by Edward Alleyn's.
Henry Condell and John Heminges compiled and edited Shakespeare's First Folio, published in 1623. Both were sharers in the Lord Chamberlain's Men from the company's earliest days around 1594 and remained with the company long after it became the King's Men, providing decades of continuity that made their preservation work possible.