Fence during new construction (Bavaria), neighbor boundaries

Accept at the border yes, but not placed on the border.

I am not very sure about that. For garages etc., the rule also applies: either directly on the boundary or at least 3 meters away. We had that case. The neighbor was instructed (not by us) because his garage was set 4.5 cm away from the boundary and should have been directly on it. Obviously, it is NOT a MUST for fences, but I still understand it to apply directly on the boundary. Bushes must be at least 50 cm away, yes.

In terms of costs, it is unfortunately somehow the case now that the one who flinches first loses. Unfortunately experienced myself, not wanting to pay but gladly wanting to decide how it should look.

Just place the fence directly in front of the boundary stone (on your side) and that's it.

We had received information from the municipality (also Bavaria) that I am not allowed to be on the border, not at the border. Garages are also not built on the border. With that, you are on the neighbor's property.

A wall from the 80s stands at our border. Meaning half a stone on our side, half a stone on the neighbor's property. The neighbor bore the costs back then as agreed. The new walls stand entirely on our side, not set back 50 cm. The neighbor had no say.

Are walls for enclosure not building walls?

But isn't it the case that one side belongs to me anyway and I am allowed to build on the boundary there?

Basically yes. But depending on the construction, you need a (strip) foundation. So it's best to clarify with the neighbor. You can't just make a foundation on their property.