How can you cool the house a bit without air conditioning?

At night we ventilate completely crosswise on the upper floor [OG]. During the day, the shutters are closed 3/4. The awning on the terrace roof blocks the sun at the terrace door, so no shutter is necessary there anymore. The photovoltaic system on the roof probably also reduces some heat. Currently, we still reach a comfortable 23 to max. 25°C inside the house. On the upper floor [OG], it tends to be somewhat higher than on the ground floor [EG]. But this is probably easier in rural areas than in the big city. Allegedly, evaporative cooling also works, e.g., hanging a wet towel in the room, etc. It just depends on the general humidity in the room.

Wet cloths help - although eventually the humidity becomes unpleasant. Damp heat is even more uncomfortable. I once had an apartment that literally turned into a sauna in the summer.

At night, we ventilate the entire upper floor crosswise. During the day, the shutters are closed three-quarters. The awning on the terrace roof blocks the sun at the terrace door, so no shutter is necessary there anymore. The PV system on the roof probably also reduces some sunlight. We currently still have a comfortable 23 to max. 25° inside the house. On the upper floor, it tends to be slightly warmer than on the ground floor. But this is probably easier in a rural area than in the big city.

Supposedly, evaporative cooling with a wet towel hung in the room also works, etc. It depends on the general humidity in the room.

so basically only a portable air conditioner with an exhaust hose works? I actually don't want that

The air conditioner cools and dehumidifies. Both are helpful in weather like today to achieve more comfortable indoor temperatures.
Evaporation has quite a decent effect with dry air. If it's muggy like it is now, you can forget about that.

Tonight we'll turn on the portable air conditioner in the sleeping area for a short time. In weather like this, it's unfortunately not enough to simply let the cool air from the forest right next to the house flow uphill through the house as usual.

We have decided against a mobile air conditioner.
If an air conditioner, then a proper one.

How can we provide some relief for now? Currently, it is not so extremely hot that we absolutely need an air conditioner.
Inside the house, it is at most 24-26 degrees.

What kind of fan would you recommend to us?
Pedestal fan, tower fan, or a fan without a rotor like a Dyson, for example?

Do you know what the Dyson can do better than other fans? The Dyson is already outrageously expensive.
We don't need air purification.

What kind of fan would you recommend to us?

The cheapest one you can still get at the hardware store.

We decided against a mobile air conditioner. if air conditioner then a proper one

Incomprehensible. Now it's warm and you don't want to cool down or can't. The 250€ for a mobile unit as a temporary solution until there's a "proper" AC is too stupid for you? Better to complain and sweat? Then buy fans that circulate the warm air. If you think you'd invest your money better that way...