Mac Pro 1,1 Ethernet not working, How do diagnose the problem
Since you don't have the original disks (and therefore don't have Apple Hardware Test), I would try downloading AHT and running it from a USB. The steps to do this are as follows:


Create a bootable USB flash drive by using Disk Utility's Erase function. Make sure you choose OS X Extended for the format and the GUID Partition Map for the scheme. For the purposes of these steps, also give your USB the name AHT when you're erasing it.
Download the AHT for your particular Mac Pro 1,1 here
Mount the downloaded image (it should mount as AppleHardwareTest)
Now you will need to copy the AHT to your USB flash drive. To start, Launch the Terminal app (usually found within the Utilities folder)
In the Terminal window, enter the following:

cp -r /Volumes/AppleHardwareTest/System /Volumes/AHT/
Now we need to make the USB flash drive bootable, so enter the following line in Terminal:

sudo bless --folder /Volumes/AHT/ --file /Volumes/AHT/System/Library/CoreServices/.diagnostics/diags.efi --label AHT
Enter your Admin password
Leave the USB flash drive plugged in and restart the Mac Pro
Immediately hold down the Option key
Select the AHT from the list and boot up


NOTE for other readers: - Both the Terminal commands and the download link used here were written for this specific question. Please do not follow these steps unless you're also wanting to download and use AHT for the Mac Pro 1,1 (i.e. these instructions will not work for other Mac Pro models).