Floor plan design single-family house approx. 230 sqm plus basement

Why do laymen think they can take on the job of an architect.

For the same reason why bad Elvis impersonators believe they "could" also do Freddie Mercury :)

Draw your floor plan on paper, preferably graph paper will also do.

Graph paper is "clearer" and easier to obtain. And don't forget: "The upper floor takes precedence" ;-)

Slope approx. 8% (east to west)
Site coverage approx. 110m^2
Floor area ratio 3 (incl. basement)
Building envelope, building line and boundary: sufficient space

For a slope, a site plan with contour lines should be available. Also, for an assessment of a floor plan that is part of a house design, the basement level is definitely missing here.

Floor area ratio 3 (incl. basement)

A floor area ratio of 3 exists, but certainly not for a 2000 sqm plot for single-family home construction.
I would say: misinterpreted.

Does the staircase need more space?

Wrong question. Is the staircase well positioned? No, because labyrinth-like corridors are formed.

Are children's rooms of 16 sqm exaggerated?

16 sqm is generally good.

In principle, you want to create 110 sqm of storage space in the basement?!
Yes, then the budget is probably not sufficient.
It doesn't help to discuss castles in the air.
With such a slope and this budget, one should stay within a two-story framework and make (a large) part of the basement into living spaces. 40 sqm is enough for technical room, utility room, and storage space.
Children's rooms can gladly tolerate a bit more light instead of dark north-facing windows, e.g. by location in the south or southwest, just mentioned generally.

I first recommend a meeting with an architect or general contractor about what the development plan allows and what the plot offers. It is still too early for a discussion here.

A floor area ratio of 3 exists, but certainly not for a 2000 sqm plot intended for single-family house development.
I would say: misinterpreted.

However, I suspect that the questionnaire – the development plan – has not even been looked at by the OP. I do not perceive any relevant restrictions here in terms of floor space index and floor area ratio (a 2k sqm plot would still leave plenty of room for a king-size double garage even with a plot ratio of 0.1 / floor area ratio of 0.15), but rather in actual building envelopes, and possibly also in the number of floors or heights. But what do Germany’s best planning software users care ;-)

But what do Germany´s Best Planungssoftware users care ;-)

The software used here is the worst the market offers (in my eyes)

The software used here is the worst that the market offers (in my opinion)

"Best" was also meant in reference to the gifted "user".

I care about the planning

So do we. Otherwise, we wouldn’t be here with you in this thread.
The planning primarily includes – even before the floor plan – the site plan. In this case with the contour lines (as repeatedly mentioned) and the building envelope.

Yes. You throw a number out there without exactly describing how you arrive at it. Please explain how you come to this value?

The number is not just thrown out. I recommend first gathering basic information about today’s house construction. Since you’re using this forum, you can also read up here.
The budget question is closely linked to the living area. If you have 400,000 or 1 million at your disposal and everyone has the same room program, the latter can plan something a bit bigger or more exclusive.
It may be that you pay less than €3,000 per sqm of living space for example at Town & Country, but you do not want to build a house from that budget provider.

Can you say what is more efficient from your point of view? That is exactly what I want to find out here

Anyone planning rooms in 2D should understand how rooms appear in 3D. At first glance, I see unnecessary hallway space and long walls with windows that are too narrow (regardless of sliding doors or dwarf windows).
I see a staircase that is located exactly in the statics of a gable roof.
From the furniture layout, I see a sofa that is not located in the relaxation/living area but moved to the dining area. If you shift it, it blocks access to the living area.
I see a dining area in the kitchen (if it is planned there) that feels connected to the public dining area.
A room that is separated by the lousy staircase position and just gets a door to the living room.

Above that a children’s room with poor spatial conditions, which also has 4 sqm more than the small children’s room.

I see that one resorts to wall/upper windows… you can do that, it’s done in row houses where you otherwise don’t get daylight.

Style: modern…

I don’t see that at all. Sight lines and more appealing spatial conditions are missing for that.

Guests per year: space for 2 persons (plan: in the study on the ground floor)

The sleeping place, which guests first reach through the living room to get to a slalom path to the toilet.

open or closed architecture (tendency to open, only the kitchen should be separable)

As said above: a large dining and living area does not make for open architecture, nor a modern construction method.

modern construction method

.

Garage, carport: planned right in front of the house (starting from the toilet to the right so that a covered path to the front door is possible)

One right, one left, north and south.
It’s best to stick to cardinal orientations.

short walking distances

Not at all. Only the bedroom is near the one shower bathroom … this also makes the bedroom a thoroughfare room, which again brings disadvantages for the residents.
It is therefore not worthwhile to make “improvements” here, as the basics like slope, basement, and statics are completely ignored, possibly also the minimum window sizes. I have no interest now in checking that.