Well, the build would easily handle streaming. But - from a video creation point of view, more cores normally == faster video conversion/rendering. Remember that your AMD build here - the CPU is a 4 core/dual module setup that does share resources. I think the shared resources does slightly impact the rendering speed, as the 6 core (3 module) FX6300 and the 8 core (4 module) FX8000 series both significantly outperform the AMD quad cores for video processing. Unfortunately both of the more powerful AMD chips are also fairly well decimated by the Intel i7's in video work. Intel setups though will cost more.
What I have seen as a solution though is putting together an LGA 1150 board with an Intel Xeon E3-1230 v3 CPU. The Xeon E3 is a quad core part, with hyperthreading, and for some reason it costs significantly less ($250 USD vs $300) than an Haswell i7-4770.
From a gaming point of view, the fact that it's a Xeon doesn't mean anything - it's still at it's heart a set of 4 Haswell physical cores with 4 logical hyperthreaded cores. You would have make certain though that the board you buy DOES support the Xeon processors. Then just get a decent one that fits in your budget - and there should be plenty of those in the $70.00 range. Since you're only going to have a single GPU, you don't need anything SLI certified, so as long as it has at least a single x16 slot (they pretty much all do) and it supports the Xeon.
If you were going to go that route:
Gigabyte GA-B85M-HD3 - $70ish
Xeon E3-1230v3 - $250ish
But it would really do everything you want, and it would do it very well.
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