] "...They called each other by their Christian name, were always arm in arm when they walked, pinned up each other's train for the dance, and were not to be divided in the set..." Does anyone have any insight into the meaning of that bolded phrase? Would they be pinning up each other’s train for dancing whilst in the ballroom, just to keep it out of the way...

I think that's exactly it, Caroline. I know I've seen prints of women's gowns having been pinned up in just such a way, but the closest I can find at the moment is in the print below, where the poor gentlewoman is having to make do with her hand, rather than with pins. The print is from a little booklet I have about fashion in Jane Austen's time. It's called: A Friviolous Distinction.

The gown shown is from 1803, and is captioned with the very phrase you've highlighted! I know that the date is five years out, but a train is a train, after all. It's quite clear to me that one could not possibly dance in such a gown without tripping-up or, horror of horrors, having one's partner stand on it!