Does anyone know why, having recorded the song in the UK in June 1964, the band decided to re-do it, with a very different arrangement, in the USA in October?

Also, Ian Stewart is credited with playing organ on the US-recorded version. Did he normally tour with the band, even on overseas trips? Would he join them on stage for the occasional number?

Yes of course he did. He was present on most tours except for a brief vacation in the 70's. Even during periods where other piano players were present Stu could often be seen at the back of the stage banging on an acoustic piano for the numbers he liked to play on. He was a FANTASTIC addition to the Stones sound and could never really be replaced in my opinion. In fact his departure from this earth really marked the beginning of the end for the Stones in many people's opinion.

Besides that he was the one who called the band to the stage when it was curtain time and honestly told them afterwards whether or not they played well that night.

In the very early days you are talking about he was even their #1 roadie, what a class act. Humble but outspoken and he could give a rat's ass about being famous,having credit or being in the spotlight. I loved everything he represented for the Rolling Stones and was very saddened by his death. peace