Fill Date field in a hook form alter with ajax callback
Here the solution finally:

https://www.drupal.org/node/1984838#comment-11736060

I had to fill both $form and $form_state values...

$form_state['input']['your_field_to_change']['und'][0]['value'] = $new_date;
$form['fyour_field_to_change']['und'][0]['#default_value']['value'] = $new_date;