ᐅ Natural flooring options suitable for underfloor heating systems

Hello everyone,
Please excuse my non-professional wording, I’m still new to this topic.
If everything goes well, we plan to start building later this year and want to gradually decide on our flooring. The house will have underfloor heating and will be occupied by 2 adults and 5 children, who often walk barefoot and have treated floors with great care over the past 11 years. I would like to keep it that way ;-). We clean by mopping after every meal and do a full cleaning once in the evening with a lightly damp vacuum mop.
We visited specialty stores and various home improvement centers. The result: more confusion than before due to contradictory information.

I’ll share our plan now and would really appreciate your opinions and expert advice:
I would like to have natural flagstone tiles glued down in the hallway and bathrooms, possibly also in the kitchen and living room. The material cost at the local specialty store is about 45 Euros per square meter (about 4.2 USD per square foot). What do you think the installation cost would be for tiles in size 30 by 60 centimeters (12 by 24 inches)?
My husband thinks the tiles are not bad overall, but they are too uneven for him. Are there smoother options available?

For the children’s rooms, bedrooms, and the office, we would like wood or cork flooring. But please something easy to install, so we can save somewhat on installation costs or be able to do it ourselves with friends’ help.
Question: Is wood or cork compatible with underfloor heating? If yes, should it be glued down or click lock installed? With an underlay? And what exactly should we be looking for? I understand that traditional planks might not be suitable for underfloor heating?
By the way, in three home improvement stores, we were given five different opinions on this ;-)
If we decide to have these floors installed professionally, what should we expect to pay per square meter (per square foot)?