The space between between frets is a mathematical calculation based on the vibrating string distance two fixed points. The instruments have different VSL's [nut to bridge]. Installing the capo on the 7th fret should work since it retains the D tuning. It will reduce the fret spacing by raising the tuning an octave [D3-A3-D4 to D4-A4-D5].
Discussions on temperament can get very involved.
Temperaments are directly related to note frequency, not note names. This is because a notes frequency, for instance 'A', can be an 435 or 445, or larger spreads. In order to set a starting point, the note frequency of 'A' is designated as the base reference point. Another part of temperament is that they define the spacing between adjacent notes in a diatonic scale. There are many more pieces to the temperaments puzzle.
A 'D4' is a 'D4' across all instruments, it may sound a bit different because of the material used to produce the note [frequency]. Just compare the note of the bass 7th fret to the open melody on a DAd tuned MD.
A D4 on an equal temperament MD may not sound/have the same frequency as a D4 on a differently tempered MD.
I have, and prefer, chromatics. The one I use the most has 23" VSL.
If you can play on a 25" VSL diatonic, you can play a 25" chromatic. The frets are the same distance apart, just some are missing on the diatonic. It should not be a problem unless you play up the scale alot, eg. frets above 10 on a 25" VSL
It would probably help understand this by comparing fret spacing between different VSL at the 'tight' end. There's really not much of a difference until the VSL's are significantly different.
This is a good calculator for comparing.
That's it on the tuning. Usually you can go higher/lower 2 half steps with no problems. A half step is one note up/down on a chromatic scale, ie; F to F#, or B to C. D to E is a full step, D-D#/Eb-E.
You almost have it! 
Instead of high to low, dAD, use low to high, DAd, it is more conventional for MD. For the other part of your question, always pluck the string just before turning the knob on the tuner. That way you know when you tune in the correct direction. As far as the F# A d, that would require a string change. You may be thinking of DF#A which needs different strings also. If you are tuned to DAA, just tune down, loosen, the middle string to F#.
DAG is Dorian, DAF is Phrygian, both will probably need restringing unless you are tuned DAA.
dAF#/DAF# is not a mode.
As an example; Standard, 28.5" VSL McSpadden tuned DAA will do exactly what you just described. Any dulcimer with the same fret layout will also meet that criteria. The caveat is the frets are really close together at the high notes.
DAA is a 155 tuning so just about any 155 tuning [w/correct strings] would work. Other modal tunings may or may not result in 2 full octaves.
DAd [158] may also work since the McSpadden has the 6.5 and 13.5 frets.
@Dusty, there are 3 octaves across all three strings, 0-6 on the bass then 0-14 [DAd] in the melody [and various combos using the middle string]. Three on one string is something else, probably huge to be usable.
I forgot to say you would need 17 frets, not counting 1/2 frets to get 2 octaves A [low] to D [high] on one string.
I'm a bit confused by all of this. Fourteen frets is two diatonic octaves, D to D or A to A or whatever, regardless of the VSL. A longer VSL [fretboard] just increases the distance between frets. It doesn't 'add' any any notes. A wider fretboard increases space between strings, but so can adjusting the middle string closer to the bass string. My recommendation would be to get something with a long VSL and adjust the string spacing between the middle and bass string. A nut set up for single or double melody strings has a wider space between these two strings built in [usually about 1/8"] by removing the inner melody string.
Happy to hear everything worked out for you.
Being new to MDs, you need to be aware that some-many vintage dulcimers have the frets located using non-equal temperaments. Those will sound good but will not sound right combined with modern equal temperament tuned instruments. You will need to know the temperament to properly tune them using an electronic tuner.
"Could they have purposely installed an actual nut to bypass the zero fret?"
Yes. The problem with that is the locations of each of the other frets is based on calculations using the VSL [Virtual String Length] from the center of the zero fret to the nut side of the saddle.
The saddle is tipping towards the nut a bit because of an oversize slot so if it still needs a bit more adjustment after fixing the zero fret try wedging a bit of credit card/toothpick/hard paper on the nut side.
You shouldn't need anything fancy. All you are trying to do is make those slots deeper so the strings are in solid contact with, or bend over, the zero fret.
I think I would try cutting the slots in the false nut deeper as a work around for now. Be sure to protect the wood.
Once you get the nut problem fixed the error should be minimal.
An example of splitting the error, Say the note/fret you check is sharp by 10 cents, you can re tune the string to 5 cents flat then when the string is pressed it will only be 5 cents sharp at that fret. The caveat is this affect all the notes/frets. If you have to fudge a bit, a little sharp is more acceptable than a little flat.
One other tip is to split the error between # and b.
First, welcome to FOTMD.
Are the strings touching the fret right next to the nut? If not, you need to cut the slots deeper because that fret is the zero fret. A zero fret performs the same function as a nut on a guitar. That black piece looks like nut but is just a string spacer.
What we haven't mentioned is higher tension requires more finger pressure.
According to the Strothers string gauge calculator, .012, .016, .024. The calculator has a tendency to be a little light [~.002 or so]. It will be the same octave, D3, A3, D4.
The ones you have on the 27" seem a little large,
https://strothers.com/string_choice.html
The most important impact of temperaments at the user level is they usually don't play well together.
I got my first MD after I was advised to learn more about HD's before trying to build one. I stopped by the Dulcimer Shop in Mt View and bought a McSpadden kit, which I still have. I decided then I needed to learn how to play it. It turned out I'm not into the typical music played on a diatonic MD so I went chromatic. I still have several diatonics although I don't play them. Ironically, I still play a lot of 'MD' music because the folks I jam with play a lot of it. 
I use several other instruments though, chording or bass. I am not a musician in any way, just want something to do!
I'm an old guy and have come to the conclusion that there are a couple of things that affect the 'apparent' demise in the use of MDs, along with a bunch of other 'skills/interests/hobbies'. The folks we see have the time and money [retirees anyone] to to invest in 'outside' interests during the 'work day', and those are the ones we see 'cause we are them. Remember, many of the last few champs are young folks, those in the work force or school.
Here's some background info. The Fender app looks like it's made for specific instruments, guitar and uke and one more and only displays the notes that pertain to those instruments. Dulcimers need a tuner that displays all of the notes, a chromatic tuner. I use gStrings [android]. It is quite accurate and has advanced features you can use as you learn more.
MD standard tuning, Bass- D3, Middle-A3, melody- A3 or d4.
The MD is a ' Modal ' based instrument, not a ' key ' based one. I think theory has 'diatonic scale' [and its naming conventions] being based on the Ionian Mode and the remaining modes being a reordering of the the Ionian mode. The nut/zero fret sets up the Mixolydian pattern. Fret three sets up the start of the Ionian pattern. Each fret sets up another mode.
This leads me to conclude that, basically:
CDEFGABC is a diatonic scale with its attendant naming conventions, and
DEFGABCD is a modal scale [Dorian] without the naming conventions. I'm wrong probably, but it keeps things simple for me. 
Addition of the 6 1/2/13 1/2 permits the Ionian mode to start at the nut/0 fret.
DddA is switching the strings from D3-A3-d4-d4 [standard DAdd] so the sizes would stay about the same, D3-.022W, A3-.012, d4- .010, adjusted to VSL.
Sanding drum[s], round files, roundnose end mills [combined with flat cutting for the longer spaces]. Use sacrifice scrap wood on each side to prevent tear outs.
I got mine from strings by mail.
There are some flatwound, nylon core strings being used by the ukulele community. I think they are used on classical guitars. Thomastic Flatwound .027 & .030 is one, there are others.
https://forum.ukuleleunderground.com
Edit:
D'Addario has some strings that may work also. NYL028W [melody on my Blue Lion bass]
I have some of their PB023/PB024 phosphor bronze I haven't tried yet.
You may need to use fingernail polish/superglue at any cut line.
First try changing the string slot or possibly the anchor point. The last thing is a single spot of glue to anchor them [may need to be replaced in the future]. A thin piece of paper/foil behind the bridge [anchor side of bridge] may tighten the slot enough to prevent the bridge from sliding.
If you retuned from DAdd, you may need to install a melody string the same size as the middle string. The melody string used for DAdd is usually smaller/lighter gauge than the middle so when it's loosened to A the tension goes down.
The test instrument is a tuner. Tighten the string until the tuner indicates you're at the correct pitch. If you use DAd it's D3-A3-d4 on the tuner or D3-A3-A3 for DAA. You stop when you reach the pitch target. This all assumes you have the correct strings for the tuning to be used.
This works because the note frequency/pitch is dependent on the tension and mass [size and material] of the string. The string calculator uses a formula to determine the appropriate size string for a frequency/pitch, the tuner reports the frequency/pitch. If an inappropriate string is used it may break or cause damage.
Since I usually do one string at a time I take it right up to the final note or just before the final note. Then I do the final fine tuning after all the strings have been changed. That takes care of floating bridges and remembering which string I'm doing. 
I've added recorders, ukuleles and micro base so I think it's more curiosity about something new [like a kid with a new toy], or exploring, or maybe just a short attention span.
It boils down to the fact that the frequencies of the notes are not exact/even and do not double [times 2] as is commonly accepted. This is the basic reason the frets are not evenly spaced and results in the 'Rule of 18' and '12th root of 2' for fret placement. I think 'wolf notes' are also part of this.
When the "G#" frequency approximately doubles, the frequency is not the same as the doubled frequency of the Ab. This difference is/can be audible.
When 'equal temperment' was developed, this difference was, essentially, eleminated.
Most of the other 'temperments' are attempts to modify this difference.
Since he's a guitar builder and probably player, I think he is referring to blues notes.That's the reason for my recommendation.
can someone help me with a fret scale chart of a mountain Dulcimer.
Preferably with the blue notes
Your best option may be to make it chromatic. The frets are basically located using the same procedures as you use making a guitar. If you decide to use the dulcimers diatonic fret placement, you will need to 'bend' notes to get the blue notes. You may want to study the MD fret board before making a final decision although the extra frets can be installed later. Or build two, one diatonic, one chromatic.
I use PDFSam Basic it's open source [free]. Be aware that you will need to go back and merge any multiple page tunes.
Since you have 2 MD, If they have the same VSL they should be the same spacing. Otherwise just use a fret calculator to give reference numbers.
Will they scrape off? The fretboard probably has no finish to damage [oiled?], so you could try using a new single sided razor blade [or fine {400} grit sandpaper on a popsicle stick edge?] and carefully scrape between the frets without marker dots.
Other than the tuneing machines, your MD really doesn't look too bad from the photos. You can do the obvious things needing fixing yourself, new strings [normal maintenance], replacing tuneing machines is pretty easy although selecting new ones is a bit harder. Just pull and replace the end pins, they're probably escutcheon nails or brads. Don't worry about the scratches/chips, it's part of having a used, probably older, instrument. I heard a pro country singer used an older guitar that actually had holes worn through the top. Don't use anything with silicone in it to clean the outside since it cannot be removed, it will prevent any refinishing or other repair work to the wood.
Bottom line, don't pay for anything more than tuneing machine installation at most.
