Is the ventilation system connected incorrectly?

Against the dry air, a moisture recovery unit could help as an additional module for the controlled residential ventilation. Some manufacturers offer this as an option.

With a smoke pen, I then examined the ventilation ducts in the house.

What do you mean by ducts? At the valves, one should be able to determine the flow direction with the bare hand without further aids (i.e., whether it is supply air or exhaust air).

The construction manager said at the beginning that air is extracted in the wet rooms (bathroom and kitchen) and fresh air is supplied to all others.

That is not wrong, but it could lead to significant overflow volume flows and correspondingly large volume flows in the bathroom and kitchen. Are there planning documents that you have?

Exhaust: Guest room (upper floor), office (upper floor), gallery (upper floor), dressing room (upper floor), utility room (ground floor).
Supply air: Bedroom (upper floor), bathroom (upper floor), bathroom (ground floor), living room 3x (ground floor), kitchen (ground floor).
Is that correct?

In my opinion, the correct would be:
Supply air: Guest room, office, bedroom, living room
Exhaust: Dressing room, utility room, bathroom, kitchen
The gallery could be understood as a corridor, usually an overflow area.

The way it is with you is deficient. I would report this as a defect. If you have no planning documents, request to inspect the documents (depending on the contract situation, you might also insist on handover, I don’t know...).

If you have the planning documents, balance the supplied and extracted volume flows.

For the time being, I would switch off the system, continue to monitor the humidity, ventilate manually as needed, and observe whether the headaches and eye redness improve (keep a diary!).

We did not apply for KFW. I will call the company.

Is it a central or decentralized ventilation system? I assume central, since otherwise not much would be feasible. To me, it sounds like someone in the utility room connected the two connections the wrong way round. Or the impeller is rotating the wrong way ^^

Even if the ventilation system is running the wrong way, hardly anything changes about the room climate. The airflow rate is theoretically almost the same whether it's wrong way or right way round. A ventilation system dries the house out, which leads to red eyes, scratchy throat, etc... You could turn the ventilation system down a bit and see if the humidity rises slightly. Normally, you can automate it and, for example, have it run stronger during shower times and turn it down for the rest of the day.

Whether air flows into or out of the room can be seen at the door. New doors are so easy to move that the draft carries the door with it. In our bathroom, the door is always pulled into the bathroom (opened). Of course, if the door is caught in the latch, nothing happens...

new doors are so easy to move that the draft carries the door along each time.

Not here, I would never have expected that in my life. Roughly 30 m³/h through an open door (0.5 m by 1.8 m) results in a speed of less than 0.01 m/s. Typically, air is also guided through a somewhat larger gap under the door and the residents can still open the door effortlessly. Your doors are obviously very easy to move!

Hmm yeah our doors also stay in the position they were left in