Frame ventilation - How large does the air gap need to be?

Yes, what else should I comment on? It is basic HVAC knowledge. If you plan closed doors, you must provide transfer openings or elements.

The airflow between the rooms in the supply air, transfer air, and exhaust air zones of a house or apartment must not be obstructed by closed interior doors. All rooms must have non-lockable and properly sized transfer openings. These passages become necessary in systems where air is supplied to the rooms from outside through outdoor air inlets (decentralized) or via supply air inlets (centralized).

How to implement these is up to the builder or the craftsman. Whether the existing door gap is sufficient can be calculated or taken from tables/diagrams.

How is it with my bedroom and dressing room? Basically supply and exhaust air in one room but with different air flows. Can I omit the frame ventilation there?

The difference between supply and exhaust air will still continue to flow…