Hi,
Too bad that your template is not compatible with bootstrap. If it would, you would have had automatically resized popups for mobile devices without doing anything.
Regarding the add to cart popup, the best is to indeed create a template override file in your template's html folder implementing the hikashop_popup_render function based on the code of the administrator/components/com_hikashop/helpers/cart.php file. That way, you don't loose your changes when you update. But if you're not a developer, then it's easier to directly edit the code of the file. It's actually quite simple. Just change the size on the line:
'<a rel="{handler: \'iframe\',size: {x: 480, y: 140}}" id="hikashop_notice_box_trigger_link" href="'.hikashop_completeLink('checkout&task=notice'.$url_itemid,true).'"></a>'.
Regarding the edit address popup during the checkout, you can do a similar change to the line:
<a id="hikashop_checkout_<?php echo $this->type;?>_address_edit_<?php echo $address->address_id; ?>" 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>
of the file "address_view" of the view "checkout" for your frontend template via the menu Display->Views.