Our servers are running with 100% uptime recently so it's definitely not the server's fault. The most obvious reason why your Host cannot connect is a firewall problem, namely blocked outgoing ports which are used by the Host.
NOTE: the article mentions version 6.5 which has not been released yet. The beta is coming this week though. In this new version the number of ports will be reduced - in fact you'll only have to have one outbound port open (5655) in addition to port 443.
You mean the Host icon in the system tray, correct?
If so, I'm afraid this cannot be possible. The icon is part of the program branding, besides it must be immediately recognizable due to security reasons and be associated with the program that runs behind it. No doubt antivirus software companies will frown upon this ability, should we start to provide it.