Short answers, pulled from the story.
Lennon wrote I Am the Walrus to confuse people who were subjecting Beatles lyrics to serious scholarly analysis. He was prompted by a letter from a student at his former school, Quarry Bank High School for Boys, explaining that their literature classes were analysing Beatles songs.
The walrus in the title comes from Carroll's 1871 poem "The Walrus and the Carpenter", from Through the Looking-Glass. Lennon later admitted he had not read the poem carefully enough and was dismayed to discover that the walrus is the villain of the story.
The BBC banned I Am the Walrus for the line "Boy, you've been a naughty girl, you've let your knickers down." The word "knickers" was the specific reason cited for the ban.
On the 29th of September 1967, Lennon tuned an AM radio to a BBC Third Programme broadcast of King Lear and fed it live into the mono mix-down. The passage used is Act IV, Scene 6, lines 219-222 and 249-262, with Mark Dignam as Gloucester, Philip Guard as Edgar, and John Bryning as Oswald.
The Mike Sammes Singers, a 16-voice choir of professional studio vocalists, performed backing vocals on the recording, contributing nonsense phrases and shrill whooping noises. Founding Moody Blues member Ray Thomas also stated in a 2015 interview that he and Mike Pinder contributed backing vocals to the session.
The song appeared as the B-side to "Hello, Goodbye" and also on the Magical Mystery Tour EP. When "Hello, Goodbye" topped the British singles chart and the Magical Mystery Tour EP reached number two in December 1967, I Am the Walrus occupied both positions at the same time.