"Reference by name" is standard DNS, you just need to use the entire <host>.<domain>
name. If you want to have .<domain>
appended automatically, go to network connection settings, look for "DNS Suffixes".
As I recall, iBook G4's had the AirPort Extreme (802.11g) card, whereas iBook G3s had the original 802.11b AirPort card. So it's good you have a G4 because that card was capable of WPA and WPA2, whereas the older card could barely do original WPA.
Note, however, that the AirPort Extreme card in your iBook G4 was only certified compliant with the original WPA, not WPA2. Apple added WPA2 support later but never got that feature certified by the Wi-Fi Alliance, so it's possible there are incompatibilities. Then again, without knowing the make, model, hardware revision, and firmware version of the 802.11 home gateway you got with your Verizon FiOS service, I can't check whether your 802.11 gateway was certified WPA or WPA2 compliant either. For example, I know that Verizon often gives people ActionTec MI424-WR home gateways when they set you up with FiOS service, and the original revision of that product wasn't certified for WPA2 either. Only "Rev B" and later are certified.
It's interesting that you say you can type your password and it never enables the OK button. The only reason I've ever seen it do this is when your password is not the minimum length of 8 characters that the WPA and WPA2 standards require. Is it possible that your Verizon FiOS 802.11 gateway accidentally let you violate the spec and set a password that's only 7 characters or less? If so, try changing your 802.11 network's WPA2 password to be 8 characters or more and see if that fixes it.
If your 802.11 home gateway is currently configure for WPA2-only (a.k.a. AES only, AES-CCMP only), you might consider configuring it for WPA2 Mixed Mode (a.k.a. "AES and TKIP", "WPA2 + WPA", etc.), or pure WPA mode (TKIP only), to see if that old AirPort Extreme card handles it better.
Best Answer
There are a handful of approaches, from the most overkill to least overkill:
I'd start with option 4. If your router supports defining multiple wireless networks with different SSIDs, then this is typically how they implement it (separate subnets, firewalled off from each other) and will likely be sufficient.
If nothing else, setting up a "guest" wireless network with a password that has no other limitations on bandwidth that your work machine connects to will likely suffice.