Re: Who brought the past Doctors to Gallifrey?
The Tardis took them all there.
Sexy doesn't always take the Doctor where he wants to go, but always where he needs to go.
More to the point, I'd say there's at least some indication that The Moment is indeed the Doctor's TARDIS from the far-future. The Moment's dialogue about confusing the past and the future felt very reminiscent of Idris in The Doctor's Wife.
Some say that he was the first choice to play the new Doctor but turned it down when he couldn't put racing stripes on the TARDIS.
All we know is, he's called The Stig.