The title pretty much says it all. How does the 6350 do in streaming video games? Will I be able to do at least 720P?
It does a pretty decent job and should handle 720P streaming just fine. Usually your internet connection is the biggest problem. If you feel it is a little slow the FX parts overclock very well so that is always an option. The 8 cores are very very good in streaming so you could always upgrade to one of those as well.
Here is the processor's benchmark in a list compared to others. As you can see, it scores very well stock, which is good indication that it shouldn't be a problem. My i7-3770K at 4.6 GHz (a very significant overclock) scores just over 10,000 points on the same software and as you could expect, handles full 1080p 60fps streaming with a breeze. I'm sure that the 6350 would be just as capable, maybe with just a little less head-room.
What I would worry about next is your internet connection. Streaming requires a pretty decent upload speed. A good bit rate would be anything between 2000-3000 kbps @ 720p which translates to 2-3mbps. You can run a speedtest to test your connection, if you get any more than that with a low ping, you should be good.
http://www.speedtest.net/my-result/3876959212
Is this good enough for internet?
And to make sure you are saying a 6350 could stream 1080 @ 60fps?
Yes, you connection looks fine, as long as it's consistent.
Also, yes, I'm sure it would have no problem with it. But as I mentioned it also depends on the streaming application you use, I would recommend OBS, it's what Logan uses for their video and it has very low overhead. It also works with about any streaming service possible.