>on other devices?
To some extent, Yes.
>E.g. see what's attached to the router,
It depends how the router is configured but almost without exception Yes, very easily because no-one, absolutely no-one but high security sites, lock down network clients to specific endpoints. ie Outgoing connections via VPN only and no access to anything else but the router.
There's even a free utility/app available called "Fing" to see what else is connected to the network so you don't have to write your own. It will give you the device, manufacturer, ip address, mac address, hostname etc. Install it on your phone and give it a try.
>.. see what sites are being visited and read the content of the sites visited - e.g. what's posted to those sites by other users?
Network adapters can be switched to what's called "promiscuous mode" where it will report every single packet it sees being transmitted irrespective of who the sender or receiver is. All you have to do then is interpret the fields in the packet.
As usual there's already an open source utility available that will do it all for you. It's called "Wireshark" and it's very, very comprehensive. I've used it quite a lot.
The difficult part is getting around the encryption so, while I would say Yes - it is "possible" to see what data is being sent and received, for all intents and purposes it is impractical.
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