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.
Blue Lion also has the 5 string Acoustic Jam bass with a 26 1/4 " VSL and 1 5/8" fretboard and Folkcraft has one with a 26" VSL with a 2" wide fretboard. I have the AJ5 Blue Lion and removed 2 strings.
As far as construction goes, I believe they about the same as a standard, just deeper for the volume needed for the lower notes. The string tension is about the same so I don't see a need for heavier construction. I've converted ones I made to bass and baritone with no problems. I even converted a regular Tennessee Music Box to sound like a standup bass for kicks. Hard on the fingers to play though and not very loud because the box is too small.
If you go to Jan Potts public page, there is a post by 'Paula' just below the photo [her thumb nail id icon] that looks like it is probably a photo of the bridge in question.
I bend them back like Strumelia and make sure the cut end is inside the tuner hole.
If the tuning is A2E3A3 [lower than DAd] or similar it's a baritone. The physical dimensions won't be an indicator.
I heard bamboo tomato stakes from a garden center are Tonkin cane. It's used for top quality split cane flyrods and those giant scaffolds you see in China.
If you take a heavy [metal] noter [I used a 5/16 x 3 1/2 " bolt] and with a light touch, you can slide the bolt/noter on the string, without the string touching the frets, and change the pitch. You need the mass of the bolt/noter to kill the strings vibration at the point of contact. I also tried a piece of 1/2" steel rod which worked the same as the bolt. I had to use a real light touch because the MD is set up for chord/melody with low set strings.
The frets then act as lines locating the the notes. For the big spaces the note is just past [bridge side] the 1/2 way point between the 2 frets.
But, since you say you had problems with dobro slides, this may not work for you.
The noter acts as a movable fret like a slide on a steel guitar or dobro. So you can actually get the notes in the large spaces as long as you stay away from the frets. One of the advantages of having a high nut/bridge. The frets act like a 'note marker' then. The string isn't stretched as much either.
I don't think the string needs to touch a fret when using a noter. If that's true, it's the answer to the tuning problem [with a bit of practice].
Last nite I happened to think that anything you can cut to size and hold onto would work, screwdriver, chunk of a broom/mop/garden tool, tablespoon, big nail in a piece of wood or file handle, etc. I suppose some would work better than others.
I used a heat gun in a home built, stove pipe oven to harden mine when I was making split cane fly rods.
Try a dobro [bottle neck] slide. Pretty much the same thing, just bigger and it could fit over your thumb/finger.
