Single-family house - approx. 160 sqm - basement - hillside location

Without having stood in such a room myself: Isn't there a risk that 5*12 m will feel "tube-like"?

My parents' house is almost the same size, and if the rooms are nicely laid out, you don't even notice it.

Isn't it higher on the valley side?

I don't know. The meaningful views and/or sections are missing.

Isn't the basement included in the calculation?

Yes. But if it only sticks out 2m and simply the ground floor is 3m and the knee wall is 2m high, then it could fit.

Is the solution given there?

No. But it says that it is only 7.0m and not 7.5m.

Attached is another sketch from the west. The height measurement starts from the intersection of the wall/natural terrain. This means that a value x between about 1m and 1.5m is not included, as the building must have a minimum distance of 5m from the street. The explanation is really bad, but I still hope it can be understood

Don't know. The meaningful views and/or sections are missing.

But. If it only sticks out 2m and simply the ground floor is 3m and the knee wall is 2m high, then it could fit.

No. But it says it's only 7.0m and not 7.5m.

Eaves height is 7.0m and ridge height 9.5m, you are right. Small typo.

My parents' house is almost exactly the same size, if the rooms are well cut you don't even notice it.

That was the sketched design.
How do you assess the digital design with the platform staircase?

Best regards

So this means that a value x between about 1m and 1.5m is not taken into account

I couldn't find any value x in your sketch. If it corresponds to my interpretation, I don't think the approval authority shares your view.

P.S.: Tell your municipality to publish the current status of the development plan. The link leads to a dead end and the search only finds a draft status from November 2019.

Anyone calculating the [Energieeinsparverordnung] will be pleased with the building envelope in the basement, as there are just so many thermal bridges until the stairwell is completely insulated,...

The measurement of the height starts at the intersection of the wall/natural terrain gradient. That means a value x between about 1m and 1.5m is not included, since the building must have a minimum distance of 5m from the street.

I do indeed not read the otherwise more usual projection onto the street-side boundary here; but I also cannot get enthusiastic about the consequence that the height could so merrily climb up the slope together with the house. At least your interpretation from the drawing, to apply it to the recessed actual basement outer wall (instead of hypothetically to the plumb line of the ground floor outer wall), I do not share. Nevertheless, the certainty of the development plan seems too close to a wishful thinking to me.