ReCAPTCHA (No CAPTCHA) connection error: Unable to Connect to ssl://www.google.com:443. Error #0: php_network_getaddresses: getaddrinfo failed: Temporary failure in name resolution - library/Zend/Http/Client/Adapter/Socket.php:235
That's unrelated to the email issue but it does indicate an issue with your server connecting with the ReCAPTCHA service.
That could be due to network or server issues.
If there are no entries in the error log for the emails then that means the emails have been handed off to the server.
At that point XenForo is no longer involved so you would need to check the mail server log to see whether they are being sent or not.
Your host should be able to help with that if you are unsure how to check the log.