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Skyrim is more CPU-limited than GPU-limited.
Any of those cards will handle Skyrim fine if you have a decent modern CPU. With a 1090T @ 3.9 GHz, 8 GB DDR3 1600, and an ATI HD 5850 1GB, I run Skyrim on Ultra settings, 4x AA, 16x AF @ 1920x1200 at a pretty consistent 60 FPS (A few of the towns drop it into the 20s/30s in certain areas, but it's at 60 FPS in dungeons, outside, etc. when it matters).
Battlefield is another story, though. It's almost entirely GPU-oriented, so the graphics card will matter more. I don't think either card is going to be ideal for High-Quality @ 1080P (my 5850 barely does High Quality @ 1920x1200 and it's faster than those cards), but they should be fine on Medium.
Have you considered spending a little more for the HD 6870/GTX 560, which are a fair amount quicker and go for about $175/$180 (before rebate) ?
As for how much power is sufficient, that will depend on what else is in your system and what your upgrade plans are for the future. A decent 500W PSU should be sufficient, but it might be a little noisy if you have say a 140W TDP processor.
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