Do you think it will run on 350W? I have read that Dell PSUs are well made and can handle more than their stated wattage. I also read that there is only one 6 pin connector available for the GPU when this card needs two. Are there any adaptors that I could get to solve this?
It ought to be able to handle a 4850 as far as size and plugs go. If you check the Dell website, you can buy it with a 5770 in there. A 4850 uses a bit more power than a 5770 though, and you've got a ton of other hardware already loaded up in there. You could probably run it, though I would be a little nervous doing it myself. I would say get a 5770 or a 5750 (5750 will perform similarly to a 4850).
It would probably work, but I'm guessing one of the reasons you want it is because of the overclocking potential demonstrated in Guru3D's article. Overclocking it even further than the factory OC is just going to further increase it's power draw.
I'd take the safer route and go with something in the 5700-series lineup instead. Not only do they perform similarly (or better) and have less power draw under load, but they also draw much less power while idle. Yeah, they cost more... But the PC was free!
Bought the below 5770 and, apart from a little wriggling, it fit into my case without any trouble. After a few weeks it is still running fine, even with games like Crysis (which is utter merde by the way!) at max settings for several hours at a time.