Alright, so I gathered my courage and took the nut off! The strings could be left in the tuning machines by loosening them enough to slide them down the side of the fretboard. I then worked all around the nut with a hobby knife, and I'm glad I did. I think there was still some lacquer on the tresholds between the nut and fretboard. The nut was also lining with the fretboard so perfectly that I suppose it was never removed since it was assembled at the factory and lacquered.
In addition, there were uneven spots of glue here and there on the nut and perhaps also some tape (?). I think the tape might have been responsible for lifting the 4th string side... Anyway, it didn't need any tapping force after carefully finding out, where the glue/tape was still holding the nut in place.
What was a bit strange was that the nut was sitting on top of a quite a) thin and b) uneven slice of the end of the fretboard. Right under the fretboard is, I think, a gap in the headstock wood, routed for the truss rod nut, so if one was to break the thin slice that's left of the fretboard, on which the nut is sitting, I don't know how the situation could be remedied without replacing the entire fretboard (which is separate from the neck).
Anyway, the surface of the fretboard slice was fairly straight, save the minor differences in the color and finish (?) at the bottom, which was a bit baffling. I would imagine it being optimal that the thing is carved (or sawn/sanded, whatever) and then left without any finish? Perhaps it's a matter of saving time in the process (and automating it?): 1) carve nut slot, 2) glue nut in place, 3) route/sand final shape of the neck/headstock/nut, 4) lacquer, instead of 1) carve nut slot, 2) mask the contact surfaces, 3) route/sand final shape of the neck/headstock, 3) lacquer, 4) remove masking, 5) glue nut in place, 6) route/sand final shape of the nut.
Anyhow, I didn't dare touch so much of the surface, it being so thin and uneven in colour even that I was afraid I'd chase myself through it finding a uniform and measurably even look. I just scratched off the parts that I thought were traces of the said tape or other obvious goo with the blunt side of my hobby knife.
Then I cut a piece of drawing paper by the edges of the nut. The paper was a bit thicker and less uniform (more porous and laminated) than regular printer paper, but at the end it still wasn't thick enough. After two retunings and measurings, I ended up adding two pieces of printer paper underneath too.
I have white, non-toxic PVA glue, but I didn't dare glue the nut on yet; it does slide if you want to, but I have to see how it holds in playing.
Anyway, the original problem seems now gone! I think the unevenness of the nut slot heights also got better, but the biggest relief is the fact that I can now lower the action to a very smooth level. The action is about 1.5mm at the 12th fret. It makes a world of difference! I don't find steel strings hard to handle at all anymore
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It is still tuned to A#6/Gm7/"high F", with the 12-15-20w-13, but I bought me a .022 plain string that I am planning to use as a G3 and then build a set of 13-18p-22-14, tuning it to dGBE ("high d"). I have to see, if the strings finally start to rise or stick in the nut slots, or if they will still work with thin strings, tuned to high g...