A player's Navi goes un-operated whenever the link between the PET and the Navi is severed. Navis can exist without an operator, but they are severely handicapped, namely the lack of battlechips. If no one is on the other side to load the battlechip data, or if there isn't a connection to the battlechip input device (PET), then the Navi never gets the data.
PETs can take calls, surf the web, send emails, etc. while maintaining a connection with the Navi, as long as the PET remains connected to the jack in port. If you pull the jack out of the port, connectivity is severed and your Navi is now without an operator. For wireless/'bluetooth' PETs, they must stay in the broadcast range to stay connected.
Correct me if I'm wrong, but you can't simply move anywhere you want and expect to remain connected. Yes there are a multitude of locations you can jack into, but you can't jump from network to network seamlessly. You have to disconnect, identify a new network, then connect to it. It's just like a wireless laptop, even if you have it set up to automatically connect to any available unsecured wireless network, there is a period where connectivity is lost.
When a Navi is un-operated, it cannot be given battlechips since those are stored only in the real world, not within a Navi's programming. They can move freely, and use their sig attacks and buster, but that's all. They are also incapable of jacking out as well, since they have no where to jack out to.
Rogan: The all-dodge thing is only really used whenever you're busting with another player, and he/she doesn't post for a long time. If that happens, the moderator 'skips' the player and moderates their navi as if he/she did nothing but dodge that turn.