Depends what you're playing. For some games the GTX460 is better, for some the HD5830 is better. The HD5830 is usually slightly cheaper. There's no real difference in the architectures, nvidia were quite late with theirs, so the GTX460 isn't really any better than the HD5800 series in that regard.
If you primarily play games that are biased towards nvidia cards (typically games like Borderlands, Metro 2033, Aliens vs Predator) then the GTX460 is likely to be your best bet (buy the 1GB version though, it's faster than the 768MB version).
If not, then the HD5830 is probably the better buy.
In the ATI case, I'd recommend buying the card from XFX, not Gigabyte. The Gigabyte card is probably fine, but could be quite noisy. In the nvidia case, there aren't really any decent partner brands any more as most of them left since the GTX470/480 were so bad, but EVGA will probably have to do.
It's important to remember that only the 1GB GTX460 competes with the HD5830, the 768MB is a slower product.

It's worth bearing in mind that typically nvidia cards have a shorter lifespan than ATI cards in terms of reliability. You've done quite well to get an 8600GTS to last this long, the few people I knew with 8600GTS cards, they've all long-since broken.

Neither, since neither are in stock. Don't expect to gain much from an overclock of nay graphics card, whoever you buy it from, they don't overclock like CPUs, and in addition, overclocking can reduce the life expectancy of a card, not because it will break prematurely, but as cards age they won't hold the overclocks they did when new.
If you want a GTX460, get this one:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814261075
If you want an HD5830, get either this one:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814150497
or, if you don't care that much about noise, this one:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814161327
The HIS HD5830 is by far the best deal of the three really.

Actually there's not a huge difference, the HD5830 is about 170W, the GTX460 is about 185W. The GTX460 is nowhere near as bad as the 470 and 480 for heat output.
The dell case is a good argument though, as they don't come with any real ventilation as standard, and nvidia cards are less tolerant to heat.