Which flooring is suitable for underfloor heating?

Go to a suitable dealer and take a look at the samples. You can also let the dog walk over them. At the same time, inquire about the prices for installation. Flooring installer is, like painter, not a very expensive trade. In case of doubt, I would rather have the floor done properly and paint myself than have a mediocre solution for the floor because it cannot be installed properly and then call in the painter.

Here, too, glued vinyl is recommended. I would really go to the floor installer / upholsterer I trust and ask for the current prices...

I just inquired with 2 floor installers from our area; one estimated 40-45€/m² and the other 40-50€/m² just for the installation costs.

For our approximately 80-82m², these costs are too high for us.

The installation costs exceed the material price of the floor.

Wow. Almost exactly two years ago, the general contractor still charged us 20€/sqm for glued installation, including his share.

I just asked two floor layers from our area; one estimated 40-45€/m² and the other 40-50€/m² as costs only for the laying effort. These are too high costs for us with our approximately 80-82m². The laying costs exceed the material price of the floor.

These are really ambitious prices compared to what we paid 4 years ago. I can understand why one would consider doing the laying work themselves. However, don't forget: floating installation always has disadvantages in terms of heat and sound.

Yes, of course there are always slight disadvantages with heat and sound.

Yes, unfortunately the high inflation affects us everywhere :/ all prices are rising.

I would say you definitely have to reckon with at least €35/m² currently just for installation.

The disadvantages of floating installation can be reduced with good sound insulation when it comes to noise, and as for heat, it shouldn't have such an extreme impact in a KfW55 house, right?

The floating installation has advantages when it comes to renewing the floor because it is much easier :D

Which floor would be best for us in terms of properties?

pleasant (barefoot or with socks)
not very loud when walking and with dog claws (but it doesn't have to be extremely quiet, just normal :) )
not too sensitive (with normal use)
suitable for dogs and children

and what do we have to pay attention to with each type of floor?

What does lacquerd sealed parquet mean? Is it then like laminate but with real wood underneath?

not very loud when walking and with dog nails

Then you should glue it. And then vinyl or parquet would be better.