Mac Pro 5.1 - Problems Powering Up (It's either dead or turning itself on)

Hello everyone.

This is my first time on here.

I am trying to set up a Mac Pro 5,1 (Mid 2010) (details below) to use for personal design projects at home and I have run into a problem with powering up that I just don't understand, any help would be very much appreciated. I have always had a limited budget so am used to setting up and working on older machines, I feel like i know what I am doing but I am certainly no expert.

Before the OS was installed, and when I only had the SSD in there, the tower powered up and turned off absolutely fine. About a week later, when I tried to install the OS, the problems started. The machine would not turn on - when I pressed the button nothing would happen, it seemed dead, no clicks, no noises at all.

Surprisingly, about an hour later, the machine just turned itself on and seemed to be fine. I soon realised however that it wouldn't turn off or restart properly. When I tried to turn it off or restart it, the displays would go black but the LEDs would stay on and the tower would just stay on - in a kind of unresponsive state (I also later realised that it would no longer wake from sleep). The only thing I could do at this point was hold the power button down, the machine and display would then turn off properly but would not turn on again when the button was pressed.

If just left, this cycle would continue. The mac would randomly turn itself on (after a fews mins to an hour) but then wouldn't wake from sleep or be able to be turned off unless the button was held and so on. I did manage to install the OS and added the extra hard drives with it like this though.

My Attempts at a Fix:

First, I reset the SMC and PRAM, this didn't work. I then replaced PRAM battery and the power unit (with a second hand but definitely working "cheap" one from eBay. Nothing changed at first. Through experimenting though, I have realised that resetting the PRAM via Terminal (sudo nvram -c) does temporarily solve all of the issues - but this only lasts about a day, then I am back to a dead machine that randomly turns itself on and won't turn off. This is at least some kind of solution, but I would love to discover why this is happening and fix it properly, I can't really get any work done like this.

Hardware:

• Mac Pro 5,1 (Mid 2010 - Westmere) - Dual 2.66GHz Six Core
• 32GB Ram (8 x 4GB DDR3 Cards)
• 4 Hards Drives - 500GB SSD (Main) and 3 x Older HDDs: 750GB (Time Machine) and 2 x 1TB
• Dual Screens - Apple 30" Cinema Display and Apple 20" Cinema Display
• Apple USB Keyboard
• Apple Magic Mouse

Software:

• Mac OS 10.6.8 (Snow Leopard)
• Adobe CS5 Design Standard

Does anyone have any idea about what may be happening here? I have never had an issue like this before and just don't understand what is wrong. I was sure replacing the power unit would work.

Any help would be very much appreciated. I hope I have explained the issue well enough without rambling on too much but please do feel free to ask for more info, if required. I am not in any rush, this isn't urgent, but I would love to learn more and solve this problem.

Thank you in advance : )