The heartburn was probably because you didn't get any stones out (they were stuck or something). If one large one is stuck, all the ones behind it are stuck too..
I had many golfball sized stones, but when I cut them open they were actually made up of lots of smaller green stones inside. They were coated in yellow "sludge" (perhaps bile or undigested olive oil). I think these probably form in the intestines since these would be too big to come out of the gallbladder or liver. The stones inside are probably the "real" gallstones.
I've tried sleeping on my stomach after a flush, and it didn't seem to make any difference. I don't think the position you're in after a flush makes any difference at all! The old-fashioned recipe (this cleanse goes back at least hundreds of years) suggested that patients should walk around all night and shouldn't lie down at all! I say do whatever you want, since it doesn't seem to matter one way or the other.