Cultivation on a "border-built" garage. What type of roof is possible?

I don't know what other information is needed.

For example, an excerpt from the development plan so that the buildable areas can be seen

the houses generally apparently do not require a building permit.

That would be very remarkable, because then there would have to be development plans for the entire municipal area. Since the state building code in NRW is currently in a state of flux, some municipalities are currently fundamentally rejecting the exemption procedure.

One probably just has to ensure that the setback areas lie on one's own property.

And is that stated in the development plan?

that if a room with a main use arises on the garage, this use is transferred to the garage and thereby would trigger setback areas (on neighboring property).

I see that differently, if the room respects the setback area. According to the planned amendment of the state building code, it is even explicitly allowed:
§6 (11)
... Buildings according to sentence 1 also remain permissible without their own setback areas and within the setback areas of a building if roof terraces, balconies, and alcoves are erected on them that maintain a distance of at least 3 m from the property boundary. In the case of roof terraces, the walkable part of the roof must be enclosed.
Of course, only if the garage is within the buildable areas, which we do not know because we do not know the development plan.

may this room then be accessible from the main house

Building Code NRW (old and new version):
§6 (11)
... 4. also if the building has access to another building.

No, that is not stated in the development plan, that's basically how it is, isn't it?

So I have created an excerpt from the development plan and marked our house and the double garage. Also all textual stipulations. In red the property boundary.

Addendum: Now with complete property boundary.

There you go, it works! The pictures show that you have many, many possibilities open to you in terms of planning and building regulations, so it is probably financial obstacles that limit your expansion dreams. You could (partially) tear down the garage and extend the house up to the minimum setback distance, you could build a one- or two-story extension for residential purposes behind the garage, you can create ample living space behind the house or simply set up a garden shed.

You should discuss your wishes with a planner, who will then also give you a rough cost estimate of what each measure will approximately cost. They will also take into account the location of existing windows, the load-bearing capacity of the garage roof, and the building regulations.

Thank you very much for your descriptions.

I would like to make a few comments regarding the possibilities.

Partially demolishing the garage makes little sense, as it is "only" 5m wide and thus you would only gain 2m, and the double garage is also needed (or could something be done to the left of the house outside the building window, e.g. carport or prefab garage?).

Behind the garage it would be the same, you could only build 2m in width there. I think the effort/cost would not be in a healthy ratio to the benefit, especially since you would then have to relocate the door that leads out to the back of the garage.

Extending to the rear is of course a possibility. The sketch of the house was not entirely correct in this respect; on the right half there is still a kind of bay window with a depth of 2m on the ground floor. On the left half is the terrace. We do not want to add on the ground floor because that would cost garden space. That would only leave building on top of this bay window on the upper floor, which would then also cover the terrace on the left. But a lot of steel beams would probably have to be used there to absorb the loads. I think that would probably be the most expensive variant of all.

The garden house is already in place, otherwise we would have even bigger space problems.

So my favored plan would be, as I said, simply to build a pitched roof on the garage to create storage space here. I calculated with the 30°C you mentioned. You then get 2.9m height at the "house edge," at a distance of 2m there is still 1.7m height. So in terms of volume, that would indeed bring a lot.

In 2 weeks I will be at the building authority and will explain the possibilities and hopefully get binding statements.

An email to an architect from the area, who was recommended to me, was sent out 1 week ago – no feedback so far.

Many thanks and best regards

or could something be done to the left of the house outside the building window, e.g. carport or prefabricated garage?

No, it is excluded according to 3.2 in the development plan. However, an open parking space would be possible. Maybe an exemption is also possible since your garage already deviates from 3.3.

on the right half on the ground floor there is still a kind of bay window with a depth of 2m.

Again, such a detail that in a personal conversation with a planner would certainly not have given rise to some thought process after just a few minutes.