Got the nut slots done.  I'd say I ended up more like 0.5 mm than 0.3 mm, but that is still pretty low.  Lower than my electric.  The grooves for the strings are a bare 1/3 diameter, I think.  But they stay in place well once she's brought up to tune.

The action all down the fingerboard has been worth the work.  At the bottom of the fingerboard it is about 5 mm for the G string and about 3.5 mm for the E.  String action is nice and low all the way down the fingerboard, very comfortable.  No buzzing or muffling, notes all sound clear. 

It will need to settle again, but scraping the paint off the shafts of the pegs and scruffing them with a bit of sandpaper hopefully will give them enough "tooth" to hold.  Time will tell on that, but they seemed to be holding well enough for that to be promising.  The D keeps slipping down to C sharp, so that one may need a bit of chalk.  The rest seem to be holding better, but it hasn't been up to tune for long, yet.

Personally, I don't think she came out looking bad.  A bit unusual, yes, but I feel the overall effect is pretty enough. 

I'm not sure at this point if I'll do more work on her now or leave be and try to play enough to get everything to settle in a bit (and get used to the strings being on a higher plane than I'm used to with my electric).  I'm not sure if I'll put the chinrest back on in any case.  But for any who thought I might not ever get her put back together, here's pics.  LOL

Thank you very much for the analysis, cdennyb!  Very interesting to see how much of a change can happen over what was really only a few days of work.  Admittedly, work of a somewhat drastic nature.  LOL

Right now, the tung oil in the fingerboard and neck is still going to be somewhat wet/plastic, and so I am considering just keeping the instrument in exactly tune and playing it often over the next few weeks so as the oil hardens it will stabilize the cellular structure of the wood in a way that will work well with the tension of being in tune and musical tones.  For those who have never messed with finishing or refinishing musical instruments, the finish may appear to be dry/hard to touch or a fingernail test after a few days.  But it is still fairly flexible and will respond to changes in stress for quite a while.  On high end instruments that have slow drying finishes based on things like walnut oil or cold pressed linseed, it can literally take months for it to mostly harden and years for the final hardening to occur.  The instrument is playable over most of that time, but kind of like new wine that has just finished fermenting and been put into bottles or whisky just put in a wooden cask, it needs time to mature and develop it's characteristics.  Tung oil is a fast drying oil, and so it takes less time.  But it can also become brittle or crack because the drying time has been artificially accelerated.  Used in as small an amount as it was here, and on an inexpensive instrument, I feel the risks aren't too high to be reasonable.  But if this was some antique (say, 40+ yrs old) instrument with a good sound to begin with, tung could be a rather poor choice and high risk.

As to the tailpiece.. Well, adjustment there might be better to put off for a bit.  I really want to replace this tailpiece.  The fine tuners are stiff and the part you turn to adjust them is very small and hard to adjust.  But also it is "alloy".  Simple ebony tailpieces are not expensive and may be a better choice for being "tuned" to smooth out the response since one can carve a bit away from the underside if needed.  Also, the current tailpiece just "feels cheap" in it's construction.  The wood of the fingerboard and neck (and the rest of the instrument, since it hasn't been played enough, being a new factory made fiddle) needs time to settle anyway.

I have also been thinking of replacing the tuning pegs and tail peg, but I am less sure of that now.  They are working well, and while the tuning pegs look a bit unusual, they do add a bit of a novel touch to the appearance. 

So, what has been gained by all this?   Much, in my opinion.  The instrument is now very playable, to the point of being inviting/intriguing.  The appearance has gone from "nice but standard looking" to "pretty but interesting".  Her voice has been smoothed out enough that she has some soft and subtle sounds to use instead of being "shouting all the time".   She can go from being about as quiet as a normal human conversational voice to quite loud with only a little change in bow pressure/speed.  "Quite loud" would be enough to easily be heard over 2 or 3 decent acoustic guitars in a jam session.  She holds tune well now, which I wouldn't usually expect a "factory fresh" instrument for at least a couple of weeks.  She is a good solid little instrument, and any "toy" or "item of decor" feel has been lost.  

On the other hand, as cdennyb pointed out, the amount of work it took does take the instrument out of "very inexpensive"/cheap range.  When you work on an instrument, you should think of the time you put into it in terms of minimum wage (at the very least).  Sure, you can gain a lot from the work as a learning experience, but time and work shouldn't ever be considered as "free" when you are considering what something like a musical instrument cost.  Value is difficult to place on musical instruments, and even more so when the instrument has been modified or something like the neck and nut tailored to what one likes to feel.  But a usual good rule of thumb would be to compare it to the rest of the line from the same maker.  I think the MV 650 is about the top of the Mendini line, I think.  Ignoring sentimental value (my violin was a gift from family), would I trade it right now for an MV 650?  No.

But I also wouldn't recommend a stock MV300 for a beginner *unless* they want to be a beginner at violin repair and modification.  LOL  Or perhaps if they really just want something nice looking as a decor item, in which case the painted fingerboard, slipping pegs, and awful action wouldn't matter.

Anyway, thanks to one and all for encouragement, ideas and support.  And a special thanks to springer.  If it hadn't been for springer's suggestion to string it up and see how it sounded before applying more finish, I would have just probably slapped more coats on and the results might well have sounded far worse for it. 

Correct, in that there is not telling ahead of time.  But it is good to check and see if you already have a bird in hand before dropping it to go and see if you can get the one in the bush.