Other people can better answer other parts of the many, many questions, but:
'Why would someone *choose* to be on the cheaper version': Typically the daytime version was the lower budget show BUT typically had returning champions. So your family could go on the nighttime Feud and win $10K in one go, or go on the daytime show and a single win only awards half that BUT could appear on way more shows total, making your max earnings significantly more than $10K. Same for nearly any other show (Wheel, Match Game, Squares, et al): A single win was more lucrative in syndication, but if you were *good* the daytime would earn you more over a run.
Network episodes in syndication: The first best example is that at least three episodes included in the first season of the syndicated 1970s LMaD are actually edited reruns of shows from the second season of the ABC primetime run. All three episodes feature noticeable cosmetic differences from any other syndicated episode that season, and clear edits to fit the shortened syndication run time. I'm not sure if its the only time it has ever happened, but it's definitely the main one I thought of.
(Plenty of network series were sold as reruns after the fact in syndication - 60s daytime Password, Joker's Wild, Card Sharks w Jim Perry, Press Your Luck as just a few examples - but almost positive that's not what you were asking about, but wanted to leave as a footnote just in case.)