Yes, you'll be perfectly fine installing that card. Your computer has a 65w processor on a 250w PSU. I see no issue whatsoever adding such a low power card by itself with no other adjustments.
As far as the card goes, it's the best you're going to do in that price range. If you bumped up the budget to around $100, I would look at the 6770. $120 would buy you a 7770 for considerably better performance.
All mentioned cards should function without a hitch on your PSU.
That will perform just fine in WoW with a few settings tweaked (mainly shadows and multisampling), and do it quite a bit better than the 6670 you were considering. It will also be compatible with your current setup.