Sorry to hear you're having such trouble.
What is in the list box is what Windows is showing as its default recording/playback device. If those check boxes are not checked, VA is not doing anything with your setup. If the check boxes are checked, VA will tell Windows to set your default audio devices when VA starts. For those reading along, most will never need to use these options as there are more flexible options available in VA for handling audio. The settings are left in there for those that use them for specific reasons (for instance, I still use them to set speakers/headset from outside VA with a command line option).
When VA starts, it links up the speech engine with the appropriate recording device and that recording device is what is used until its told not to do so. There is no polling going on to see if that device is still available or has gone away, as that would be checking for something that relatively never occurs.
In your case, I'm not sure what is going on with your headset, as I would call it extremely atypical for a device to switch off upon mute. I have not seen or heard of any device doing that, unless the device is faulty and reports to Windows that it has become disabled or disconnected (for some weird reason). You will want to try reinstalling your device drivers and/or find out what would make your headset disconnect mid-session.