Short answers, pulled from the story.
Back in the U.S.S.R. was written by Paul McCartney and credited to the Lennon-McCartney partnership. McCartney began writing it as "I'm Backing the UK", inspired by the "I'm Backing Britain" campaign, and developed the song while the Beatles were in India in early 1968.
On the 22nd of August 1968, Starr walked out of EMI Studios after growing frustrated with McCartney's criticism of his drumming and the tense atmosphere during the White Album sessions. The other three Beatles completed the basic track and created a composite drum track without him; Starr returned to the group on the 4th of September.
The song parodies Chuck Berry's 1959 hit "Back in the U.S.A." and the Beach Boys' "California Girls". Mike Love of the Beach Boys, who was studying at the same meditation retreat in Rishikesh, suggested that the bridge section focus on Russian girls in the style of "California Girls".
Yes. Released three months after the Soviet invasion of Czechoslovakia, the song drew criticism from both the American political right and the New Left. The John Birch Society cited it as evidence of the Beatles' supposed pro-Soviet sympathies, while some on the left criticised the band for a gesture they considered tactless given the invasion.
McCartney performed the song in Moscow's Red Square in May 2003 on his Back in the World tour. He had been refused permission to perform in the USSR during the 1980s. According to The Moscow Times, the crowd went wild.
When issued as a single by Parlophone on the 25th of June 1976 to promote the Rock 'n' Roll Music compilation, the song peaked at number 19 on the UK Singles Chart, number 11 in Ireland, and number 19 in Sweden. Billy Joel's live cover version reached number 33 in Australia and number 44 in New Zealand.