Site becomes inaccessible due to PHP-FPM listen queue, CPU touches 100%

Try stopping your NewRelic agent and waiting a few hours to see if that resolves the issue. If it does, then try upgrading it to the latest version. If it comes back once it's upgraded, contact NewRelic support.

Check the max_execution_time and request_terminate_timeout in your php.ini.

Check the proxy_connect_timeout, proxy_send_timeout, proxy_read_timeout, and send_timeout values in the Nginx config as well.

I would recommend checking your TCP/IP settings to see what those are as keep alive and time out settings there may need to be reduced. I've seen some distros come with a minute or more by default.

You should also verify that the traffic from the listener is valid traffic. See if you can out put samples to a file and validate that the traffic is legit. Many automated processes seek out Wordpress instances on the interwebz. These bots can cause all kinds of problems as they to hack your site.

Have you check your access.log or domain.com.access.log at /var/log/nginx/? Looking to that you will have more detail why PHP-FPM eating your CPU.

I think your website is on a brute-force to wp-login.php, that's consume a lot of CPU.