0. I seriously doubt the option "New User Account Activation" is set to "Self" at the time of this writing. I registered in checkout, and I was immediately registered, activated, logged in and able to continue. So, no activation email to be expected.
2. Registration is not a problem, works perfectly fine without hinderance in checkout. No other registration form I can find on your site, so no way of testing other ways. (You should probably create a menu item for Joomla or HikaShop registration at least for testing purposes.)
Your Joomla template does not display system messages. Proof: there's seemingly no shipping method configured, so I can't proceed in checkout, and the usual error message in this case is not shown. So, error messages in case of emails failing to be sent are not shown, either.
Like with registration, there seems also no other way on the site for me to test general mail functionality, such as a Joomla contact form or else. It could well be that the hosting server refuses "email relay" if you use emails of a different domain than the domain in use with the host.
Sorry, but under all these conditions there's no more one can say from the distance.
Bottom line: You need to verify and possibly adjust/correct your settings for Joomla and HikaShop. You should see that your Joomla template displays system messages (check with your template provider); meanwhile you may check the various logs for errors. And you should test Joomla (and HikaShop) registration and emails with native on-board functionalities/menu items to be able to verify they are working.