Sounds like the machine isn't getting to logic; CPU, RAM or logic board issues most likely.
The biggest trouble with G5 machines is failing solder connections, most commonly around memory controllers and slots.
This is commonly acompanied by failed POST at startup with the machine beeping or power LED flashing.
However, the same solder connects all other components to their respective attachments; CPU connections to the card could also be at issue.
Solder issues are very often sporadic, especially at first, often allowing just enough operation to give a glimmer of hope to a fatal condition.
Sourcing a copy of the ASD 2.5.8 would give a nice troubleshooting test suite.
Reseat the hard drive connectors.
Try booting to OS X Install Disc.
Remove the RAM (noting position), reseat, try starting.
(I would) then reseat the processors.
Reseated the ram and made sure everything was in place, booted fine from the PCI card but still no go on the AGP card.
I have tried a flashed Nvidia 6200 card, it didn't work properly as the rom was for a 256mb card and this one was only 128mb but it did produce a distorted display that told me that the AGP slot is working, I was beginning to wonder if that was duff.
so...it's staying with the PCI video until I can find another AGP card to try.
Found a AGP card in my old gigabit G4 (Rage something or other 32 mg) but it had a lug on the front that prevented it going into the G5's AGP slot. On closer inspection the lug was not electrically connected to the rest of the card, it just looked like it was there for support.
So.. one cut off lug later I popped it into the G5 and pressed the power button and the machine boots the desktop totally happy with the 'new card'. I now know the AGP slot is fine just hit on unlucky with 2 broken graphics cards.
It will do for now