Progress:
It seems there is no problem with compressor and gas after all... because heating works OK, after pressing "Heat" button and short period of time spend in stand-by mode, compressor started and the temperature of the grill reached 40°C in couple of minutes - so unit seems to be mechanically and electrically sort of OK (last weekend I didn't wait long enough).
However, the control board (thermostat?) may have the problem because compressor definitely doesn't start in cooling mode.
Interestingly, some time ago someone re-made the front panel for some reason as home project, see attached pic (not too bad work though).
I really need wiring diagram now
