Alright, I'm not exactly sure how intense your video editing is, but you could probably step down to an i5 3570k and over clock it, which should provide you with enough processing power for simple rendering/edits, and offers very similar performance to an i7 3770(k) in-game. If you were to over clock, get an AsRock Z77 Extreme4 board, which has a very reasonable price. Otherwise, the rest of your build is great
In gaming, not really. It all depends on the extent of your rendering and streaming. I agree, the 3770 will increase performance, and perhaps that's what you're after, and in that case if I were you, I would probably just get a 3770k and over clock the heck out of it. An OC'ed 3570k vs a stock 3770 will probably give higher FPS in games to the 3570k, rendering more towards the 3770 with the extra threads. With a build in your budget spectrum, overclocking should definitely something to consider, and so a K processor and a Z77 motherboard are something you should look into. i5 3570k/i7 3770k is your choice, sorry I wasn't completely clear in my last post.