rivatuner gives you details about the pipes and whatnot on the card.. Haven't used the program in a while, so I dont remember exactly. If you were to like overclock alot, I guess it's almost kind of plausable that a single part of the gpu, a single pipe or whatnot could become damaged, leaving you with a card that can sort of function sometimes. You may want to consider testing out a very old driver set. Newer drivers utilize more features of the card and work them harder sometimes. May be worth a shot. I'd also look into rivatuner to see if you can gain some insight through there. Like I said, I haven't used it in a while, and I'm far from an expert, but somebody may know more.
another thing.. It'd be a lot of work, but if you have time, you may want to try reseating the card.. If it wasn't in the slot correctly, it probably wouldn't boot, but you never know.. There's a thousand things that could cause crazy behaviour.. a tiny piece of conductive dust could really throw everything off. Regardless of how illogical it seems, I would still try it, just in case.
with the AC thing, I was just using tha as an example. It didn't make any sense, but the bios flash fixed it.
At this point, your original assumption that the card is fried seems about right, but I wouldn't give up yet