All fields are cleared, after pressing save button.

https://www.youtube.com/watch?v=dfqttimOIYU&feature=youtu.be
?

I had exactly the same issue, finally decided to ditch this plugin and use a workaround. WordPress already has a jQuery datepicker, you just need to run the script. I put this right above the closing head tag.

<?php wp_enqueue_script( 'jquery-ui-datepicker' ); ?>
<?php wp_enqueue_style('jquery-style', 'https://ajax.googleapis.com/ajax/libs/jqueryui/1.8.2/themes/smoothness/jquery-ui.css'); ?>
<script src="https://raw.github.com/jquery/jquery-ui/master/ui/i18n/jquery.ui.datepicker-ru.js"></script>
<script type="text/javascript">
      jQuery(document).ready(function(){
         jQuery('#your-textfield-id').datepicker({
            dateFormat : 'D, d/m/yy'
         });
      });
</script>

Do not use this plugin – you will lose forms if you press save.
This is a serious bug that will destroy your existing forms.

(Copy a form. then save it to see what it does!).

Thanks arekMP,

Your workaround is working really well.
I always prefer to use as less plug ins as possible in my sites.

However your date picker is Russian.
English version below

<?php wp_enqueue_script( 'jquery-ui-datepicker' ); ?>
<?php wp_enqueue_style('jquery-style', 'https://ajax.googleapis.com/ajax/libs/jqueryui/1.8.2/themes/smoothness/jquery-ui.css'); ?>
<script src="https://raw.github.com/jquery/jquery-ui/master/ui/i18n/jquery.ui.datepicker-en.js"></script>
<script type="text/javascript">
      jQuery(document).ready(function(){
         jQuery('#your-textfield-id').datepicker({
            dateFormat : 'D, d/m/yy'
         });
      });
</script>

There is a conflick on your site – I tried to disable all plugins and my theme to default – CF 7 worked! Dont blame the plugin first!?