MacOS 10.8 Mountain Lion can't connect to external services (App Store, iLok, Avid, etc)

Hello there,

I have an old Mac Pro 4.1 with MacOS 10.8 Mountain Lion installed. It is a fresh installation - it was an old system and I wanted to make a hard clean-up of it. I thought this would fix the issue as described in the title, but it still persists. Namely - I can't connect to any app-related network service. What I mean by that - I have Internet connectivity, the browser works just fine. However, all the programs such as iTunes, App Store, and most importantly: iLok Manager and Pro Tools - all throw "server unreachable" messages (in Pro Tools it's where I am trying to browse anything related to my Avid account).

My network setup is just plain router and Ethernet cables; still, I have checked the router's configuration to make sure. I have also checked another networks such as mobile hotspot from my phone, all with the same result. My guess is that I have some setting missing perhaps? Or network component set up in some old way. I tried all the fixes found on the Internet, such as clearing caches or databases of App Store, some proxy settings in the Internet options, but without any luck.

I would highly appreciate all the help - I have spent quite a few hours trying to fix this by myself, which only ended up with so high frustration...