It is a little unclear what you have done. I've been using 5970's for a couple of years and have had six GPUs fail out of my farm in that time (different models).
It looks like you have applied paste to the top of the GPUs, like you would with a CPU install. Do you then attach the big, ugly heatsink? If not, when the card starts the chips will heat very quickly and turn off (thermal trip) so you might not have killed it yet.
The back-plate is a combination of protection and heat dissipation - the HD58xx series doesn't have them. While playing with other pieces, it's ok to leave off.
Can you confirm you have a heatsink attached/bolted to the GPUs and at least one fan running?
I would NEVER run any GPU or CPU without thermal compound and the heatsinks mounted properly, not even for a few seconds, unless it's an ancient GPU that didn't come with any cooler from the factory.
I did attach the big heatsink as well and I made sure that proper contact was made between the GPUs and the heatsink. Upon doing a cold boot, just after I push the power button to start the computer I can hear the GPU fan spin up very fast for about 2 seconds before it spins down again.
Does anyone here know if anyone will take the cooler off the card if I return it to the shop it was bought from? I'm thinking I might have to put some thermal pads between the stock heatsink and card so it looks like it hasn't been tampered with. I just don't know if the people at the shop, or if the supplier will take the cooler off and see that the thermal pads have been replaced with thermal paste making it obvious that someone has been playing around with it.
