The 6870 is definitely the better card. If it is only $10 more, go with that one. It performs better than the 560 in BF3 but will also perform better in other games if you will play other games too.
You can overclock any GPU with a decent cooler on it. Read the reviews and see what users say about their temps with the coolers from different manufacturers.
As for the PSU, go with Corsair. Much more reliable brand and the Gaming Series provides good amperage on the 12V rail.
Let me know if you have more questions