Setback distances for new construction near an external garage?

An extension on such a narrow house sideways at the rear part adjoining it is unlikely to become a role model either in terms of beauty or traffic area efficiency. The micro-location "neighbor refuses a win-win land swap" also does not speak in favor of a passionate desire to stay. I do not see the resale value of a house wrapped around by two garages being able to exceed the construction costs of the extension. From my point of view, therefore, there is nothing against the alternative of building new elsewhere. This will hardly realize any greater loss on balance. As horrendously unattractive as I find the property just based on the Tetris constellation in combination with the house "width," there is still a demand pull for that even in Munich / Stuttgart / Hamburg. I therefore clearly tend towards "object change before extension."

Do you not have to provide a parking space or could you possibly buy your way out of it? If the answer is no to both, the question would be where to put the parking space with a new house... instead of a basement, an underground garage?

And do you absolutely want the house to be usable separately as such, or would an extension also be sufficient? In case of needing care, a single-story would be preferable anyway. About 8x5m around the two garages, see sketch. A separate entrance door at the bottom of the plan with a garden gate (because presumably no driveway will be allowed there) with its own east terrace or a shared garden in the west. A connecting door to the existing building and the upper floor of the extension you could either use yourself or you could build a platform lift from the open living room on the ground floor upstairs with bedroom and bathroom.

An architect would be the most sensible from now on.