Hi,
1. First, I would recommend going to the Customers>Emails history menu of HikaShop. There, you can see all the emails sent by HikaShop, including the activation emails.
If you see the emails there, then HikaShop sent the email to the email sending server of your hosting for delivery. If it was not delivered, the reason will be logged in the log file of the email server and your hosting provider should be able to tell it to you.
If you don't see the emails there, then it means that the "user account" email has been deactivated in the System>Emails menu of HikaShop, so please check there.
2. HikaShop uses Joomla messaging system to display the message. It's not HikaShop which displays the message but your Joomla template, which takes it from Joomla messaging system.
The default template of Joomla, and most Joomla templates out there, don't display the messages as popups.
I would recommend you to temporarily switch to the default template of Joomla and try with it. You'll see that the message displays normally.
So, if you see the messages as popups, that's something your template is doing and you should check with your template provider what they recommend.