Reconstruction by neighbors along the property boundary

Thank you for your perspectives, I will look all of that up and communicate it. Thank you

Why is your house located so close to the border?

are you sure about that? In my opinion, if at all, the land charge is entered in the neighbor's land register or rather in a land charge register of the municipality.

I will quote the state building regulations.

"(7) 1In the setback areas as well as without their own setback areas, even if they are not built on the property boundary, the following are permissible
1.
Garages including their ancillary rooms, covered underground garage driveways, elevators to underground garages, and buildings without habitable rooms and fireplaces with an average wall height of up to 3 m and a total length per property boundary of 9 m, whereby the height of roofs with a slope of more than 45 degrees is added to one third, with a slope of more than 70 degrees fully added to the wall height; gable surfaces remain disregarded for a roof pitch of up to 45 degrees,
2.
building-independent solar systems with a height of up to 3 m and a total length per property boundary of 9 m,
3.
retaining walls and closed fences in commercial and industrial areas, outside these building areas with a height of up to 2 m.
2The length of the construction not complying with the setback depth to the property boundaries according to numbers 1 and 2 may not exceed a total of 15 m on a property."

Accordingly, the neighbor’s project is legal if there are no further structures on the boundary.

Here is the regulation on setback distances.

"(5) 1The depth of the setback areas is 0.4 H, in commercial and industrial areas 0.2 H, but in each case at least 3 m. 2Through urban land-use statutes or a statute according to Art. 81, a different depth measure of the setback area may be permitted or prescribed. 3For such regulations in zoning plans, § 33 of the Building Code applies accordingly."

Evening, so in Bavaria there is no Baulast, as I heard, and everything is recorded in the land register. Why it is as it is, I don't know. It is my mother's house and we rented there in the 90s until she bought it in 2001. I went to the municipality and got floor plans, it is marked in red. I am still working on the exact measurements and will share them (the above ones are not entirely wrong yet). Now there is slowly conflict. She wants to continue building after a verbal agreement and is already 90% finished anyway, but mother is now backing out because of the gutter agreement that extends over our property and is obstructing or is getting legal advice. Unfortunately, the appointment is only on 17.4. and the neighbor's roofer wants to finish it on Friday.

As already said: the burden should be registered with the neighbor. According to the wiki, in section II of the land register.
Because there, the neighbor would have the burden due to your construction being too close. So far, so good.
However, you will not be able to look into it. And it means, in turn, that she must keep the boundary clear, even if it is on her property.

The gutter should virtually not play a role, because what she is building there is, in my opinion, not allowed. Even if cites the state building code, it can also mean something else.
She must adhere to the construction burden. Of course, I do not know the exact details. But if it is as I think, her shed is illegal. Possibly her old shed still had legal protection, but it has been demolished, so she would need a permit for her shed.
How to proceed now?
I would probably – also because fire protection is no longer observed, nor privacy, and probably not laws either – go to the building authority again with this drawing. In principle, what you have drawn is sufficient. They also have the exact measurements in their cadastre. I would avoid the neighbor for the time being until you have more clarity regarding your rights.