Hi,
From what you're saying it seems to be a problem with the SEF on your website.
If you turn off the SEF in the joomla configuration, the issues should not happen anymore.
So that means that you need to add an &Itemid=XXX parameter in the link of the email to avoid the SEF system to display a 404.
You can edit the emails via the menu system>Emails to do that. So for example, for the order status notification email, you would have to replace the line:
$order_url = $data->order_url;
to:
$order_url = $data->order_url.'&Itemid=XXX';
with XXX being an id of a public HikaShop menu item and the links of confirmation/shipping notifications should then work properly with the SEF turned on.
For the user account settings, it's because you need to enter a menu id of a joomla menu item of the type "joomla user profile" in the "menu" option of the hikashop user account plugin and that will then work properly.