Expand the house or a crazy idea

Or can you still add a room under the roof?

Why does everyone assume that the older child moves out at 18/19? It is totally uncommon here, unless they go to university.

I would bring a structural engineer, an architect on board, preferably those involved, and ask what is even possible.

Since the lines to the utility room run through the carport and they are not allowed to be built over.

Then you take 1m² from the new dressing room for a house connection room and then lay the lines above the extension without involving the supplier until the existing utility room. When the shell construction is finished, the supplier disconnects the lines in front of the new floor slab and reconnects them in the prepared new utility room. Your plumber then only has to connect the new lines to the old connections.
You can easily get the necessary expansion of the building encumbrance from the neighbor? Otherwise, you could also maintain the 3m boundary distance on the west side. The new room would be big enough anyway.

Then you nibble off a 1m² house connection room from the new dressing room and then lay the lines above the extension to the existing HAR without involving the supplier. When the shell construction is finished, the supplier will disconnect the lines in front of the new floor slab and reconnect them in the prepared new HAR. Your installer then only has to connect the new lines with the old connections.
You can get the necessary extension of the building encumbrance from the neighbor without problems? Otherwise, you could also maintain the 3m boundary distance in the west. The new room would be big enough anyway.

Which new dressing room are you talking about? The lines now come through the carport on the lower house wall via a multi-service house entry into the house. This side of the utility room (so on the floor plan then the north side) is completely full of lines. Water/gas/electricity/heating....

The building encumbrance to build up to 2 meters to the boundary is already registered with the neighbor. Or do you need a new one for this...? The building encumbrance was registered and only then the house including carport garage was planned... so I assumed that this applies to the entire building plot... They couldn’t have known whether we build 1 meter or 20 meters...

By the way, I am not necessarily in favor of expanding and think it's fine as it is with the children... but my better half sees it differently... although the additional rooms wouldn’t be unsuitable for me either. With 5 people you really need a lot of space... now that they are small it’s still okay. But then when the teenage years start...

Is there nothing more to do in the attic?

If I remember correctly, it is only 150cm high - at most, a sleeping niche could be installed above the children's rooms.