Room cold at the feet - Is basement ceiling insulation useful?

Not really. You can only access that side if you knock out the old screed and lay something underneath. 2cm is not even enough for PUR, then you need vacuum insulation. Very expensive - but also very effective. And yes, don’t skip ceiling insulation, that causes thermal bridges again.

If you lay parquet or instead of parquet a wool carpet, it will also be less cold on the feet.

If you lay parquet or instead of parquet a wool carpet, it will also be less cold underfoot.

Should that be a mediocre joke? There is even a carpet on the parquet. Of course, it is just as cold as the parquet. Anything else would be surprising.

A question arises regarding implementation: In this room (storage room) there are large wooden shelves that reach up to the ceiling and are anchored there. I suppose that should be changed and the panels should not be glued around them?

Basically cover the entire ceiling. If it is only the studs/posts of the shelves, you can leave them out. That is only a few square centimeters, which does not matter. And since it is on the cold side, there are no problems with moisture.

Regarding the living room: If the floor covering is ever renewed (currently a somewhat aged parquet),

Insulating does not help in my knowledge, since wood does not conduct heat, but only assumes the ambient temperature.
The constant topic "parquet despite underfloor heating" exists, but it is not excluded because there is parquet that is "suitable" for underfloor heating, since the thickness of the parquet makes it possible.
You now have older, probably thick parquet, and it simply takes your ambient temperature from the room. At least, it should be so. It does not conduct cold from below or not noticeably, and thus the cellar ceiling insulation will not help either. Of course, I have not experienced this personally, I am only concluding.

Even after hours of heating (normal radiator) one still feels chilly due to permanently cold feet - the floor then has a temperature between 17 and 18°C

How warm is the room? Not at the radiator, but centrally a) at the floor b) at 1.5 meters height c) under the ceiling? Heat is known to rise...

Is that supposed to be a mediocre joke?

Honestly, I would and still would point out that floors in old houses without underfloor heating (or even without a cellar) are better covered with carpets - not without reason did people used to have Persians and runners everywhere.
Floors without underfloor heating simply feel less cozy.
I would suggest that you have a new floor covering laid and underneath it a mobile electric heating coil / retrofit underfloor heating or similar.

According to my knowledge, insulation does not help because wood does not conduct heat, it only assumes the ambient temperature.

This is effectively the same, since this ambient temperature is not static but there is a constant heat transfer occurring between "above" and "below". Thus, the average between the temperatures on its warmer and colder side is constantly changing. Insulation is basically nothing more than a very high impedance.