Garden house and WEG shared corridor parcel

The problem seems to be rather that 2.5x2.5 m with the wish for 30m³ becomes about 3x4 m ... inner dimension of a brick hut., outer dimension therefore about 3.4 x 4.4 m = about 15 m² compared to 6.25 m² before. And with very small plots ... .. how big is the total plot and how big is the garden area you are talking about?

The plot on which the house stands is about 260 m² + (plus my share of common areas about 100 m²)
Total about 360 m² plot.

The ground area of the house is just under 60 m² --> I therefore have about 200 m² garden without common areas.

A distance of 3 m to the boundary is maintained for my garden house.
Another special feature is a water protection area 3. Surface infiltration and basin infiltration is only permitted.

Any idea how I can best get a permit?

Is it your own property where you want to place the garden house, or [Gemeinschaftseigentum mit Sondernutzung]? In the latter case, any structural change requires the approval of the WEG.

The building permit for the overall concept of your developer for the condominium-divided parcel with stipulations also for the private roads, paths, and parking spaces for cars was granted in 2017 with various deviation regulations that maximize the buildability of this area.

Something there is indeed, according to statements, pushed to the limit.
Possibly it is the floor area ratio?
From here it is somewhat guessing.

The city responded to an inquiry we made:

What kind of inquiry was it? By phone, in person, in writing? And regarding content, with concrete plans, specific building descriptions, or general, like the second paragraph of your opening post?

How do I best proceed to get my small garden house?

You ask your developer to send copies of the approved building documents and the building permit including all conditions, exemptions, and deviations. Alternatively, you can request these from the building authority (for a fee). Then ask someone who knows about this (surveyor, architect, civil engineer) whether what you plan is approvable. For non-binding advice, you can also upload all the documents here, at least the development plan and a site plan.

Is it your own property where you want to place the garden house, or is it common property with special use? In the latter case, a structural change requires the approval of the homeowners' association.

Actually, everything is common property.
However, there is an agreement that one can build/change on the property without the approval of the entire residential area.

Something is definitely maxed out according to what was said.
Possibly it is the floor area ratio?
From here it is a bit of guessing.

As far as I can see, the floor area ratio is quite maxed out. But I also wonder how a garden house counts into this.

Then ask someone who knows about this (surveyor, architect, civil engineer) whether what you plan is approvable.

Who would be best to ask here? ;)

As far as I see, the floor area ratio is pretty much exhausted. However, I also wonder how a garden house counts towards this.

A garden house counts towards the floor area ratio.

Again: if you do not provide any concrete information here apart from shaky terms like "pretty much exhausted," "actually," etc., no one can help you.

Otherwise, everything you should/must do is stated here, i.e., what the city will advise you:

Alterations to the building measures are always assessed and decided with regard to this overall complex and therefore also require a building application with building documents according to the Building Documents Ordinance, prepared by an authorized draftsman according to §67 Hamburg Building Code."