<<< The XP PC will only let me get access to the W2000 PC if I disable the firewall for the latter. >>>
It is the firewall in the Windows 2000 PC that is preventing you from accessing, so you need to enter the IP addresses of the networked PCs in Zone Alarm's "Trusted Zone".
<<< The W2000 PC will not let me get access to the XP PC, even if I disable the firewall for the latter. It prompts me for the XP password, but I haven't set one up! >>>
By default, File & Printer Sharing is disabled in XP. Have you actually set any folders as "shared" in their Properties, because doing so will enable F & P Sharing. The request for a non-existent password is usually because F & P Sharing is still disabled. However, also have a look at this page - click here
I would also suggest you install the same free Zone Alarm in XP and disable Windows' own firewall which is only a "one-way" firewall and doesn't control outgoing internet traffic.