The CC has a maximum of only 10 MB RAM. A lean System 7.5 would work, but I stronlgy suggest to stay with System 7.1 with a few selected Extensions like Thread-Manager.
The CC can run OpenTransport and there should be no problems with Networking. I did this in the early 2000s when I also had a CC with network card.
Regarding the webbrowser. The most "modern" one you can get is an old version of iCab Look for v2.99 for 68k Machines. Beware, it won't be any fun browsing the modern web with a machine like that. It more or less just does not work. There is a small but slowl