Obviously you have a problem with the video card. This is one of the things I hate most in working on laptop - the need for a temporary replacement for analysis
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If you can find some video cards to use, I suggest you try replacing both just for testing. The 'very hot to the touch' card, if you ask my opinion is defective thus the high temp. Start with replacing the two and see if you can at least go and see your BIOS settings either on the LCD or on an external monitor.
Post back what you find out.