Everything you said points to a memory issue. Have you tried installing the RAM modules one at a time in slot 1 (closest to CPU)?
- Power off, unplug
- Press/hold power button for ~15 sec
- Remove all RAM except the one in slot 1
- Remove motherboard battery
- Press/hold power button for ~30 sec
- Reinstall the battery (right-side-up!)
- See if it boots now with only mouse, monitor and keyboard connected
- Repeat the process, except swap all of the other modules into slot 1 until you identify a bad modiule
- If they all work in slot 1, repeat the process but this time add a second module to slot 2, etc. until you've reinstalled all of them or identified a bad slot