Underfloor heating vs. room height, what should one do?

The current floor construction is 6cm.

According to the cross-sections, 6cm without further definition. In the construction certificate it says covering linoleum (but now there is parquet and I don't know what else) 2cm, subfloor screed 5cm, sound insulation (or thermal insulation?) coconut fibers 2.5cm

The controlled residential ventilation system can be hidden in walls and the intermediate ceiling beneath the screed, underfloor heating, and the rest.

On the upper floor, there is still the attic above, and on the ground floor, the basement is below. Would it be a possibility to lay the pipes there and then lead them into the rooms via floor/ceiling openings? However, these areas are not heated. Insulation around the pipes? Risk of thermal bridges?

According to the cross-sections 6cm without exact definition. The building permit states covering linoleum (but now there is parquet and I don't know what else) 2cm, subfloor screed 5cm, sound insulation (or thermal insulation?) coconut fibers 2.5cm

You are still considering whether to buy the house, right? If the rough details fit and you then move into this detailed planning (underfloor heating, floor etc.), I would definitely have a thorough survey carried out. Measure new floor plans, check the floor in all rooms, etc. That can cost a few thousand €, but then you have a basis for further planning without having to constantly rush into the house to check something. Apart from the fact that the new plans should also be drawn with as realistic measurements as possible ;-)

The controlled residential ventilation should remain within the heated envelope. Otherwise, you inevitably bring problems into the house.

Sure, you can insulate everything, but then you need more space again and have even higher costs.

The engineer made us the following proposal (after I said, without underfloor heating is not ;)): Schlüter-Bekotec, installation height 52mm. Does anyone know this? It's a pity that he didn’t offer this right away. However, he can't say anything about the costs.

Measure inside the house first? Are you the only interested parties for the house? I'm surprised that you have so much time to calmly plan and obtain offers.

As long as the statics is not a problem, everything else should be solvable; the others have already described how to do it.