My question is: are we allowed to build a terrace roof covering the entire property (i.e. 9m wide and 3m deep) without a building permit?
The limits for permit exemption in NRW are 30m² in area and 4.50m in depth (Building Code NRW, §62, paragraph 1, g). Your roof covering is therefore exempt from the permit. Whether you are also allowed to build it is a different matter, e.g. in §6 Building Code NRW, any existing development plan, or any existing design statute. If there is a development plan and your terrace lies within the designated building area there and closed construction is prescribed there, and there are no other relevant statutes for the building plot, you can simply build. In case of open or deviating construction, you need at least the consent of the neighbors, which must be reported to the permitting authority. They will then check whether the consents meet the public law requirements. In some cases, building encumbrances are required. If the terrace lies outside the building area and the development plan does not explicitly allow this, neither terrace nor roof covering is permitted there, and you must submit a waiver application to the permitting authority, which must be well justified. As you write, the house is still under construction, so you should ask your designer for advice, as he should know the local building regulations very well.