I have a hunch that you have at least two problems, probably a CPU tray that is misbehaving and a corrupt BootROM.
If this was happening to me with a Mac Pro, I'd test each component top to bottom with a known working Mac Pro, but seems you don't have one right now to eliminate each defective component.
Maybe you can borrow a working Mac Pro from a friend or buy one on the cheap from a recycler. I'd do that, buying a working Mac Pro is a lot cheaper than the parts to repair one, obviously you need to keep your eyes open, takes sometime to find the really good offers.
I'd stop for now, without having a Mac Pro to test each component you will gonna have a tough time to find what is/are the real problem (probably more than one, my bet). Get your Mac minis working, while doing it you can try to find a cheap Mac Pro to diagnose yours.