Question about the 6 1/2 fret
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I doubt it. But that's a good question.
Yes. Absolutely. You cannot play tunes based on the mixolydian mode otherwise. So "Going to Boston" and "Old Joe Clark, " for two common examples, necessitate a 1-5-8 tuning. My guess is that people referred to the tunings by common tunes. So 1-5-8 might have been referred to as "the Old Joe Clark tuning" and 1-5-7 might have been "the Shady Grove tuning."
More generally, I think you are right to connect full-length frets with extra frets. The 6.5 fret allows the 1-5-8 tuning to get the major 7th note of the major scale, but melody notes below the tonic have to be played on the middle string. So the 6.5 fret alone would not necessarily allow a drone player to play in the ionian mode. (Not trying to scare anyone with fancy terms, plagal melodies require using the middle string in 1-5-8 but authentic melodies do not.) My point is merely that only if we are fretting across the strings can we make full use of a 6.5 fret.
Having said that, some drone-style players do indeed make use of extra frets. Don Pedi has both a 1+ and a 6+ on his Modern Mountain Dulcimer dulcimer, although he often uses more traditional dulcimers for demonstrations.