After I update the information on the new TC following your instruction using the Airport Utility, can I unplug the second, newer TC, move it to another room and plug it in to another RG45 plug?
Yes. When you transfer the settings, AirPort Utility will reset the old TC back to factory default settings, so it can be set up anyway that you want.
Can I set up a separate network for the newer TC?
The newer TC will have the same network name and password as the old TC, since you are asking about transferring settings from the old TC to the new TC.
Use the setup wizard on the old TC to set it up to extend the network from the new TC. Both TCs will operate at their maximum speeds this way, and you won't have to change networks when you move a device from one area near one TC to another area near the other TC.
You can have each TC provide its own network name if you want, but you'll have to change networks whenever you move a device from one area to another. That's quite a hassle for most users who don't want to have to log off of one network and log onto another when they move a device from one area to another.
Get the new TC setup and working first, then we'll tackle the older TC setup.