Connecting your machine right to the router via Ethernet usually works fine, at least in my experience. I‘be heard that there are some modern routers and switches that sometimes can’t properly detect and connect to older, slower Ethernet ports… not something I’ve ever experienced with my FW SE Clamshell (as I typically have an older Ethernet switch for just that purpose), but could be a possibility.
Does that Ethernet cable work fine with other machines? Maybe try a different, known good cable, or try that cable with other machines to confirm that it works.
A universal OS9 install disk should have installed all the needed extensions for wired networking at the very least. I would also assume that a universal installer for OSX should have installed such things as well.
My gut feeling on the wired part is a compatibility issue between older and newer hardware, or maybe just a bad cable. Might end up being software as well, but that is unlikely, though it depends on exactly what install media and options were used.
As for wireless, modern wireless routers can also have issues with the original AirPort’s wireless b and the older WEP/WPA security. Sure, you can disable such security on your main router, but I wouldn’t recommend doing that long-term. I know some newer routers do not allow older wireless b/g devices to connect unless you specifically turn that feature on, but again, that’s not something I would leave turned on long term.
Usually best to have an older AirPort or other wireless b/g router that you only power on and connect such machines to when needed, and leave unplugged when you’re not using it.
If you’re just transferring files over to the machine, you can always just sneaker-net stuff onto it with an external USB drive or a USB thumb drive. Kinda sucks at USB1 speeds, but it will work for that purpose, though slowly. That would save you the headache of having to deal with such wired and wireless issues, but doesn’t really fix the specific problems you were asking about.