The issues re: the security certificates may be a result of your browser needing to update it's root CA
. Check with your browser vendor on how to do that.
WRT "no internet connection"...again, that's a browser/ISP connectivity issue, not something related to this site. A web server won't return a status code of "no internet connection" (the closest is maybe a 404 not found). It's your browser, OS, or your connection to your ISP that is having problems.
I connect regularly from a heavily firewalled and isolated corporate network with no issue, as well as over 3G and 4G cellular connections, in addition to my WiFi connection at home (via a 75Mbps Verizon FiOS connection) on my tablet, laptop, and cell phone, all with no issues.