Hi,
I invite you to look at the file administrator/components/com_hikashop/views/order/tmpl/listing.php
You'll find this line of code:
echo JHTML::_('calendar', hikashop_getDate((@$this->pageInfo->filter->filter_start?@$this->pageInfo->filter->filter_start:''),'%Y-%m-%d'), 'filter_start','period_start',hikashop_getDateFormat('%d %B %Y'),array('size'=>'10','onchange'=>'document.adminForm.submit();', 'onChange'=>'document.adminForm.submit();'));
As you can see there, we call JHTML::_('calandar', ...) to display the date selector for the filter of the orders listing. This is the standard Joomla calendar selector.
So we're already using the "default joomla calendar everywhere a calendar is needed".
The problem is elsewhere and more complex than you make it seems.