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It seems, I found some solution, may be not fully drupal-way.

Firstly, we need to load JS libraries, needed for working of ajax submit. This code helps us:

drupal_add_library('system', 'drupal.ajax');
drupal_add_library('system', 'drupal.form');
drupal_add_library('system', 'jquery.form');
drupal_add_library('system', 'druapl.progress');
drupal_add_library('system', 'drupal.textarea');

Nextly, we need to fill JS class Drupal.settings.ajax with data about page elements, which provides ajax behavior. I have not an universal solution, I only stops JS code on breakpoint in misc/ajax.js when running it on page with correct working ajax form. You can see correct values in this part of code:

Drupal.behaviors.AJAX = {
  attach: function (context, settings) {
    // Load all Ajax behaviors specified in the settings.
    for (var base in settings.ajax) {
      if (!$('#' + base + '.ajax-processed').length) {
        var element_settings = settings.ajax[base];

        if (typeof element_settings.selector == 'undefined') {
          element_settings.selector = '#' + base;
        }
        $(element_settings.selector).each(function () {
          element_settings.element = this;
          Drupal.ajax[base] = new Drupal.ajax(base, this, element_settings);
        });

        $('#' + base).addClass('ajax-processed');
      }
    }

So, in my case resulting JS code, needed to be added into startup javascript, looks like this:

if(Drupal.settings.ajax == undefined){
    Drupal.settings.ajax = {
        'edit-submit': {
            callback: "faqform_ajaxsubmit",
            event: "mousedown",
            keypress: true,
            prevent: 'key',
            submit: {
                _triggering_element_name: "op",
                _triggering_element_value: "Send"
            },
            url: 'system/ajax',
            wrapper:'submit-message'
        }
    };
}

If all done correctly, you form, loaded by ajax, will have correct ajax behavior, in my case - ajax submit.

May be there are more graceful solutions, may be diving deeper into code will help you to find them.