well what you said about the 15 steps i have noticed some of what you said only applies for systems like NT, 2000, XP or 2003.
with the turning on by itself after 10-15 mins of being shut down that's just weird
so you've got the problem tracked down to the power supply? maybe it would be worth replacing it - it shouldn't be too hard to replace it as long as you know how to plug the new one up