Short answers, pulled from the story.
"Flying" is credited to all four members of the Beatles: George Harrison, John Lennon, Paul McCartney, and Ringo Starr, with the writing credit listed as "Harrison/Lennon-McCartney/Starkey". It is the only Beatles track officially credited to all four members.
The basic track was recorded on the 8th of September 1967, with tape loop overdubs added on the 28th of September 1967 by John Lennon and Ringo Starr. It first appeared on the 1967 Magical Mystery Tour release.
Lennon plays Mellotron and contributes sound effects, McCartney plays bass, Harrison plays guitars, and Starr plays drums and maracas. All four also contribute wordless vocals, singing the melody without any lyrics.
The tape loop overdubs extended "Flying" to 9 minutes 38 seconds, but the track was edited down to 2 minutes 17 seconds for official release. A bootleg version featuring the original jazz coda runs approximately 2 minutes 8 seconds.
In the Magical Mystery Tour film, "Flying" is accompanied by colour-altered images of Icelandic landscape filmed from an aeroplane, as well as unused footage from the 1964 Stanley Kubrick film Dr. Strangelove.
The Mellotron's Dixieland Rhythm setting is one of the instrument's banks of a pop orchestra playing popular musical styles, with optional accompaniment. It was used to perform the jazz-influenced coda that appears on bootleg versions of "Flying" but was removed from the official release.