In console, we can see error that relates to one of your plugins http://prntscr.com/k7ep75 Please try to disable all 3rd-party plugins and then check the registration.
Also, ask your host to check mail logs to find the reason why your site doesn’t send emails. They should find some errors if there are any.
Sorry… me again.. I just want to make sure that I’m not missing anything… For the registration form to be filled in and for the person to receive details about their new account… Am I right that I should have a pluggin such as Form 7.