When you modernized your heating system at your house from oil to air pump, was it possible to reuse existing floor heating? Or usually oil system with water distribution to floor heating is not compatible with air pump system?

That’s exactly what we did when we converted our electric boiler to geothermal. Can’t imagine an airpump would be any different.

Our system also provides hot water !

Changed from gas to air pump which heats the water in our underfloor heating, not a problem.
We have had to have the underfloor tubing purged since, but that's because of the age of the fluid in the pipes, not because of the heat source.

For us, that was part of the price.

The problem only became apparent a few years later, I didn't think of asking the company to flush the system at the time.
Looking back I should've done..... hindsight is a wonderful thing!

I am just wondering what is typical distance between heating plumbs when connected to oil heating system? I think oil heating works on higher temperatures of liquid circulating in the plumbs than air pump? So with oil heating the distance might be larger (lower density of plumbs), while air pump there might need to be closer to each other. Wasn't it a problem when you were switching from other systems to air pump?

Floor heating is 35C and capped at 55C regardless of heating source. Oil likely could produce 100C (or more) and that's why there is a "mixer" and "preventer" are installed for such system. And very same when you have multiple independent circuits.

Distance is not any issues as mostly you'd have separate set of circulation pumps and which are sized for overall resistance and volume transfer of water.

Is it possible to have an oil burner drive underfloor heating in some rooms and regular (old and heavy) radiators in other rooms?

...yes if installing an heat exchanger with radiators as primary loop and underfloor heating with separate pump as secondary loop...