Floor plan of a two-family house, ground floor and attic apartment

That's a start. :) Just to clarify again: 1 child, 2 adults upstairs? 2 adults downstairs? Do you share the utility room? A garage directly attached to the house is not desired? If so, why not? Such a passage is just a wind tunnel and you sit behind the house in the wind.

So, the parents at the bottom. And upstairs then possibly later a family with a max. of 2 children. Currently no child yet, but planned for later :-)

Yes, the utility room would then have to be used together. The washing machine should go in there.
If the upper floor is rented out (you never know what might happen later), we would then have to plan a connection for the washing machine in the kitchen.

The garage is supposed to be a prefabricated garage. A connection to the house is probably not that easy or certainly costs extra. Besides, the space between the garage and the boundary would then be dead space.
And we would like to enter the garage from the back with bicycles and a motor scooter. (5m for the cars and 2m for bicycles, etc.).
And one could possibly build a roofed walkway from the garage to the entrance later on.
That's the idea.

Regarding the situation: In front of the house is a street (being newly built). (...) 3 m from the street is not buildable and then the building boundary of 15 m starts to the rear.

Is the street then located to the east? Please check the building boundary again with § 6 Abs. 5 ThüBauO, accordingly (H is the wall height according to Abs. 4): (5) The depth of the setback areas is 0.4 H, at least 3 m So it could also be more than the 3 m...

In front of the exterior walls of residential buildings of building classes 1 and 2 with no more than three above-ground floors, a depth of 3 m for the distance area is sufficient.

Only the 3 m apply, right?

3 m away from the street are specified in the development plan.
And then we have 15 m to build on.
We have planned a house measuring 10 x 13.5 m.
Garage 6 m wide, leaving 1 m between the house and garage.
In front of the house space for parking spaces (4.5 m) and then the house with 13.5 m. 1.5 m + 13.5 m fits within the building line of 15 m.

That was our idea.

It concerns the plot with the arrow.

I see the problem that you have a higher space requirement upstairs for 4 people than downstairs. Downstairs is for 2 people. They need less space, but possibly already age-appropriate, if not even barrier-free. That a technical room can be accessed by both parties is not unusual. However, there then needs to be a storage room upstairs. How about a balcony? The family upstairs would probably also like to get some fresh air now and then? I would simply create a room program: which rooms are approximately needed in which sizes? Is the plot suitable for a two-family house (BePlan)?

Yes, exactly, the ground floor should be as barrier-free as possible and less space is needed downstairs than upstairs. That is the dilemma. A balcony and storage room would be nice. But at least the balcony exceeds the budget. However, you can always add it later. Otherwise, you can set up 2 terraces. The room layout corresponds to my design. And the more space, the higher the costs. We are modest after all :-) Yes, 2 residential units are allowed.