I just wasted 4 hours on this problem and figured i might aswell document it, so that if someone else is going to run into the same issue, they won’t have to suffer the same 4 hours that i just did.
If you use AIR Applications under Windows and all of sudden those apps can’t connect to the internet anymore, make sure that your Internet Explorer is not “Working Offline”. You can check in the Menu of Internet Explorer under Tools -> Work Offline. If it has a checkmark, remove it and your apps should be working fine again.
I’m really suprised that the Windows Version of AIR apparently retrieves it’s Network Capabitilies via Internet Explorer and that this behaviour is in no way documented anywhere. Hopefully Adobe will changes this or atleast make sure that people now about this.