Get the i7 now. You're already spending out, you don't want to buy it twice just to toss (or try to resell) the i5 later. A GPU is also MUCH easier to install down the line, and for the games you're looking at the iGPU should work fine for a while.
I'd say that a 5820k is going to be the best of the three for livestreaming, while a 6700k will be better for gaming (more cores that streaming can fully use, vs better single-core performance that most games will use).
Also, don't get lost chasing numbers. There's a big difference between *playing* at 60fps vs 30fps, and streaming at 60 vs 30. Streaming at 60 is mostly a waste of bandwidth you really can't spare, assuming you're streaming to Twitch and aren't a Partnered caster.
The only reason to do so is if you're playing a retro game that uses sprite blitting to fake transparency (Super Metroid, Shovel Knight, etc). At that point, sprites either stay solid or disappear entirely on-stream due to capture sync rates.
Even then, you generally need to drop to 480p@60fps to fit within the advised 2000kbps maximum, instead of the 720p@30fps 'golden point'.
Which isn't to say that an i7 still won't help you; it will. You'll be able to run a lower x264 preset, which will improve the image quality of your stream through better compression at a given bitrate.