Hello, me again.
I just wanted to give a update on this issue, since i've been continuing to tackle it. I really think this is a driver issue that is neglected by Apple. I recently abandoned macOS on the Mac Pro 3,1 since it is not supported beyond El Capitan, and installed Windows 10 straight up without using boot camp on the 2nd internal disk using a Windows 10 DVD. Well guess what, the ethernet ports are working just fine and dandy for well over a month now. I realize that this is probably not a solution for most people here but i thought i would point out that discovery. For my purposes, all the apps i need to use are also available for windows. On the hardware side, everything seems to work perfectly on the Mac Pro with Windows 10 except the built in speaker and built in bluetooth, but for my purposes as a headless server, that's a non-issue. I much prefer the stability.
What i had been doing before installing windows 10 is i had setup an automator script that would make the ethernet ports inactive and active again, twice a day. My specific issue would get resolved simply by going into system preferences and making the afflicted ethernet interface inactive, then active again. So with an automator script running on the calendar app every 12 hours it was as close as i could get to having consistent network uptime. But then came the time that the ethernet port failed soon after the script had run, and there was a downtime of almost 12 hours. So to **** with the workaround and hacks, i decided to give Windows 10 on the Mac Pro a try as i found a few articles online saying it worked well. I'll give it a few more weeks and then i'll wipe the 1st disk with macOS on it and use the space for Windows.
I used to have more macOS machines running in the house than Windows, this install just turned the tide.