The fx will play them, but it will not be able to handle all games in the future without stuttering.
Perhaps one of these would suit you better? http://www.tomshardware.com/reviews/gaming-cpu-review-overclock,3106.html
Since the cpu is pretty old you would have to look at ebay and similar places.
There are some x58 motherboards in your price range, but finding one with sata 3 is not as easy.
Sometimes the screws on the cooler are not tightened properly resulting in bad contact of the thermal pads. Try tightening them but be careful not to break any seals or you might void the guarantee.
You might want to wait a month or so for AMDs nano gpu, as it it might offer better bang for buck than the 970, or at the very least drop the price on it.
The Nehalems are only that low on the chart because of the much lower stock clocks.
A properly overclocked Nehalem still all but smashes a piledriver in single threaded performance, which means it's still to this day more relevant the any fx cpu.