When you go on your address popup edition, I can see that in the form, the Itemid is lost because it's not passed by the URL of the edition link.
And since you have the "force menu on checkout" option activated, after the redirect to the checkout, the step is lost when redirecting to the correct menu.
So I think that the solution will be to add the itemid parameter to the edition URL. For that, you can change the code:
<a title="<?php echo JText::_('HIKA_EDIT'); ?>" class="modal hikashop_checkout_<?php echo $this->type;?>_address_edit" rel="{handler: 'iframe', size: {x: 450, y: 480}}" href="<?php echo hikashop_completeLink('address&task=edit&redirect=checkout&address_id='.$address->address_id.'&step='.$this->step.'&type='.$this->type,true);?>" onclick="return hikashopEditAddress(this,<?php echo (int)$same_address?>,false);"><img alt="<?php echo JText::_('HIKA_EDIT'); ?>" src="<?php echo HIKASHOP_IMAGES; ?>edit.png" border="0" /></a>
to:
<?php
global $Itemid;
$url_itemid='';
if(!empty($Itemid)){
$url_itemid='&Itemid='.$Itemid;
} ?>
<a title="<?php echo JText::_('HIKA_EDIT'); ?>" class="modal hikashop_checkout_<?php echo $this->type;?>_address_edit" rel="{handler: 'iframe', size: {x: 450, y: 480}}" href="<?php echo hikashop_completeLink('address&task=edit&redirect=checkout&address_id='.$address->address_id.'&step='.$this->step.'&type='.$this->type.$url_itemid,true);?>" onclick="return hikashopEditAddress(this,<?php echo (int)$same_address?>,false);"><img alt="<?php echo JText::_('HIKA_EDIT'); ?>" src="<?php echo HIKASHOP_IMAGES; ?>edit.png" border="0" /></a>
in the file "address_view" of the view "checkout" that you can edit via Display->Views.