"What is typical environment influence (when device is moved to another environment; assuming no big metal objects nearby in all cases)? Is it possible to avoid tuning (e.g. by changing of reference frequency)? Is it less than 1pF?"
If you set the Theremin on a table, or if there are large conductors nearby, or you take the top off the wooden cabinet, you'll probably see a ~1pF change. You could put a small variable C at the antenna, but that will likely give you stability and adjustment problems. IMO the most straightforward path here is to somehow finely vary the drive frequency via software. Close the loop for auto-calibration, then open the loop to allow touch-up by the user. Does processor PWM give you fine control (~1ppm) over the frequency?
It goes without saying, but you want to do the bare minimum in unstable analog circuitry, and do everything you can in the much more stable digital / software realm.
"What caps are you using? All my caps have 50V working range. Could there be any problems if higher voltage is applied? Antenna voltage can easy exceed 300-500V amplitude according to LTSpice model."
Ha! I'm currently torturing the 1pF 50V caps at my antennas with hundreds of volts. An ESD event would probably push this over the edge (though my plate antennas are pretty well insulated by the plastic boxes). I intend to replace these poor caps at some point with an equivalent value of 100V caps in series.
Also, voltage swing in simulation is usually highly exaggerated by Q and lack of radiation in the model. Measure what you really get on the bench and adjust Rant, usually around 1meg to 1.5meg, to give roughly the same swing. Don't use DCR of the coils in your simulations, a rule-of-thumb here is to multiply DCR by 2 or so to crudely factor in skin effect and similar parasitics.
If you go this phase route you may actually need to lower the Q by increasing the drive resistance. For your coils you can use the finest wire you can easily handle without breaking or giving you trouble when winding it. I've found 34AWG isn't too bad to handle. Get the stuff with insulation that melts with a soldering iron.