It appears to me that making a Network makes one a hub of sorts.

How CPU / BW will this be?

And if one makes such a network, is it portable?  Can it be moved to another system, another client, another location?

And if a channel is created, with many users, does the channel stay intact of the channel owner/network owner goes offline?

Networks have very low bandwidth, CPU, and RAM requirements.

File indexes are cached to disk when a user is offline, so when they re-connect there is very little bandwidth for the connection setup.  Only changes are transmitted.

Searches have been pretty heavily optimized, so even if you have a substantial number of users on your network, doing a lot of searching, CPU usage is pretty mild.

You can easily move a network to another computer.  You just need to move the root key.  There is a cache of the offline user file listings that can be found in your appdata/hosted_networks folder, but it's generally not worth the trouble of moving it since users will just re-sync the next time they connect, and even for a huge collection of files it's not much bandwidth overall.

To answer your last question, chat rooms are hosted by the owner, so the owner does need to stay online to keep it running.  You can have multiple owners and switch off hosting duties which helps a little.