Can a layperson dare to handle the baseboards themselves?

Jigsaw plus 45-degree guide, small inaccuracies concealed by white acrylic.

We thought of such MDF moldings (there are plenty available on the internet), they look white but are not real wood. I like the Hamburger or Old Berlin profile best. For damp rooms, we would then have to buy special moldings, I assume?

If you have a local sawmill that is on your side, you can have such moldings cut from MDF panels and have one edge lightly processed. That’s how I did it once. But it only pays off for larger quantities. An electric miter saw is not too expensive and can often be used in the house anyway. You can then practice nicely and quickly gain routine. And if you ever mismeasure, not much happens.

We are getting MD moldings that are also suitable for damp rooms. As long as they are not completely submerged in water, I don't worry. It gets tricky when the room corner is not exactly 90 degrees or the walls are not that straight. Have you already thought about how you want to attach the moldings? For nails, you need the appropriate machine; I don't know if there are clips. If I remember correctly, we once glued them. There is white tape for that, which can also compensate for unevenness.

Have you already thought about how you want to attach the moldings? For nails you need the appropriate machine; I don't know how clips work.

Glue and fill gaps with acrylic, as has already been written here.

Yes, I also thought gluing was the easiest. Since we're on the topic, is mounting tape better, or is it preferable to glue liquid?

There is nothing easier. But please do not glue! Use clips and drill them into the wall. The screed will settle and then you can simply correct.