How to troubleshoot ERR_CONNECTION_TIMED_OUT for my site?

What is the correct way to troubleshoot error from the title? It's happeing randomly: sometimes site works flawlessly, and sometimes you get the "Coecting..." text in browser's status bar when trying to reach it. Usually after it timeouts (half a minute) and Chrome shows the error, it will recoect blazingly fast (without hitting refresh) and open the site. When it's working, it's working really fast. Also, sometimes there is trouble SSHing to the box with Putty, coection timeouts. Again, when it is coected, it will not break, it's very stable.

It's a VM (Linode) in Frankfurt DC, OS is CentOS 7, updated yesterday. It runs a Laravel (PHP framework) application on PHP 7. Http server is nginx 1.10 and it uses PHP-FPM. Both are configured to use UNIX socket. There is no load yet on the server. CPU is under 0.5%, there is ample free memory and disk space. Network is managed by network service and eth0 script, not NetworkManager. Static address. Firewalld is down and SELinux is disabled until I sort it out so we can rule firewall out. I'm starting to think that that is the problem because I've ran out of options :) IPv4 DNS resolution has precedence over IPv6. DNS server 1 is 8.8.8.8 and 2, 3 and 4 are Linode's own.

I've set up Pingdom to check it every 5 minutes. In the last two hours it had 3 outages.

When I check it from time to time on www.downforeveryoneorjustme.com sometimes it says it's up, and sometimes down.

Where to start, which log files to analyze first? How does the browser coect to the site, what is the exact process? When it establishes coection, it works, I need to make sure it doesn't timeout.