Is it possible to check if action from some action set exists?

The simplest way is to run the action. If it throws an exception, check to see if it's the exception for a missing action.

document = app.open(new File(inputFile));

try {
    app.doAction(actionToRun, actionSetToRun);
} catch (e) {
   // Check e to see if it's the missing action exception.
}

Or you can use this function extracted from xtools/xlib/stdlib.js.

function hasAction(atn, atnSet) {
  function cTID(s) { return app.charIDToTypeID(s); };
  var asetDesc;
  var rc = false;
  var i = 1;
  var asMatches = [];
  while (true) {
    var ref = new ActionReference();
    ref.putIndex(cTID("ASet"), i);
    var desc;
    try {
      desc = executeActionGet(ref);
    } catch (e) {
      break;    // all done
    }
    if (desc.hasKey(cTID("Nm  ")) &&
        desc.getString(cTID("Nm  ")) == atnSet) {
      asetDesc = desc;
      asMatches.push({ index: i, desc: desc});
      //break;
    }
    i++;
  }
  if (asMatches.length == 0) {
    return false;
  }
  for (var i = 0; i < asMatches.length; i++) {
    var asmatch = asMatches;
    var asetIndex = asmatch.index;
    asetDesc = asmatch.desc;

    if (!asetDesc.hasKey(cTID("NmbC"))) {
      continue;
    }
    var max = asetDesc.getInteger(cTID("NmbC"));
    for (var j = 1; j <= max; j++) {
      var ref = new ActionReference();
      ref.putIndex(cTID("Actn"), j);           // Action index
      ref.putIndex(cTID("ASet"), asetIndex);   // ActionSet index
      var desc;
      try {
        desc = executeActionGet(ref);
      } catch (e) {
        break;   // all done
      }
      if (desc.hasKey(cTID("Nm  ")) &&
          desc.getString(cTID("Nm  ")) == atn) {
        return true;
      }
    }
  }
  return rc;
};