http://pcpartpicker.com/p/FJ8s7P\
Here's what I have came up with. It is true the i5 will not perform as well with video editing as the i7, which is due to its lack of hyperthreading.
I choose the i7 4790k for gaming and video editing, it will do great for that as well as streaming.
I choose the hyper 212 evo because it is the best budget cpu cooler and it will do a decent job at keeping your i7 cool.
16 gb of ram is pretty much a no brainer with gaming and video editing.
I choose the motherboard based off of its strong reliability and popularity. I know many people who have it and its done a fantastic job for them.
I was able to squeeze in a 120 gb ssd as well as a 1tb hdd. I'm dead set on having an ssd, they're remarkably fast.
I went with the evga 970 4gb ssc acx 2.0 graphics card. It will do great at 1080p gaming. And is a great card in general.
I chose the nzxt s340, Its a very nice case and in my opinion it looks really sleek.
But I think this is one heck of a build for $1200. You really can't get much better parts for this price, and you have everything that you wanted in this build.