Yeah, there's a bunch of stuff about up and downscaling which is frankly testing my limits of technical knowledge severely. But the short answer is that via HDMI, I'm not able to do the 1300X766, or at least not without only having a refresh rate of 30hz, which is (as you might expect) obnoxious and not pleasant to look at. Since my Samsung (which is also an HDTV, btw) has got a native resolution of 1300X766, and looks great with VGA, something must be up with the HDMI setup specifically. It's probably not a major deal, since apparently I got the non-Displayport 5570, and both my Samsung monitors were not Displayport in any case--so I would have had to drop more money for adapters, which seems a bit silly to me just so I can geek out with three monitors instead of two.
Given that I now have a relatively cool running card (51C idle, about 65C at full load, all with the fan speed running at about 27% maximum) compared to what I used to have (65C idle, about 83C at full load, fan running 100%), and it's obviously a better card than my old one, I probably shouldn't complain, and it seems to be working well now. But I swear, the next time I upgrade, I'll run so far away from any kind of "slimline/mini/low form factor" case... "
P.S. Any suggestions about a program which will allow me to control the fan speed on the card? I could even up that a bit, say to 35 or 40%, and see if that drops the temperatures even further...