I'm positive it's the graphics card because the machine freezes within minutes every time I power it on with this card. If I put in a different graphics card, the problem never happens. Also, I think the melted plastic on the fan casing and obviously overheated inductor are good indicators.
I do have an oscilloscope and a multimeter. However, I just discovered that this card still has 3 months of warranty left on it. I thought it was just 1 year, but it's actually 2, so I'm going to see if EVGA will replace it for me. From what I can tell, they should.
Thanks again, Alex, and everyone else, for all the help. This is one of the best online communities I've seen and it's guys like you that make it so good.