in my experience, an ill fitting heatsink would not show the symptoms you reported.
you describe CPU and case fans turning on for a second then stopping when you try to start the PC.
if the heatsink was not connected properly, I would have thought the PC would start as it should, but when the temperature got to a preset limit, the PC would shutdown, but we would be talking a couple of minutes at least.
but, computers being computers, I couldn't 100% say the heatsink may not be the root cause, just doubtful.
it does however highlight your experience level in assembling PC's.
and there is absolutely no malice intended by that.
shoot, my very first build, I dropped the socket 775 processor and bent 3 of the pins (some fine metal work with tweezers fixed it and she purred like a kitten once installed)
I'm saying, we have all been there and "you don't know what you don't know" - you won't know that part A is not connected properly, simply because you haven't done it before.
how about attaching some screen shots of the innards and we'll see if anything sticks out?