can you start a Terminal on your system and gather the content of the file:
It should look like this:
(BEGIN - you won't see this)
** network-manager - network connection manager
** The Network Manager daemon manages the system's network connections,
** automatically switching between the best available.**
description "network connection manager"
start on (local-filesystems
and started dbus)
stop on stopping dbus
(END you won't see this)
One thing that could be affecting the issue is the state of your Wireless connection itself - the icon could be sowing that your connection is no longer available (meaning that the icon is changing its state to reveal that the connection is down).
If the connection icon is a bright icon against a dark background and the icon is dimmed when the connection is not available, this may account for the 'disappearance' of the icon, when actually it is there, but the connection may have become unavailable.
A simple test would be to try to connect to any website when the icon has disappeared. If no connections are possible - then the network connection is possibly unavailable. A ping test to a local network resource that allows same would confirm that connection availability and from there an internet ping test would confirm any outside connectivity.
Give that a try and see if this helps explain the mystery of the disappearing icon.
As for opening the Edit connections, you should be able to do this if you provide the sudo credentials as required. Does this not work, or is this not being requested?