PC-Build for Gaming,Recording and Editing

Refined your build.

PCPartPicker part list / Price breakdown by merchant

CPU: AMD RYZEN 7 1700X 3.4GHz 8-Core Processor ($399.99 @ B&H)
CPU Cooler: Noctua NH-D15S 82.5 CFM CPU Cooler ($79.99 @ Newegg)
Motherboard: Asus PRIME X370-PRO ATX AM4 Motherboard ($170.00 @ Amazon)
Memory: Corsair Vengeance LPX 32GB (2 x 16GB) DDR4-3000 Memory ($238.97 @ Jet)
Storage: Samsung 850 EVO-Series 500GB 2.5" Solid State Drive ($159.99 @ Jet)
Storage: Hitachi Deskstar NAS 4TB 3.5" 7200RPM Internal Hard Drive ($149.99 @ Newegg)
Video Card: MSI GeForce GTX 1060 6GB 6GB ARMOR OCV1 Video Card ($253.98 @ Newegg)
Case: Corsair Carbide 400C ATX Mid Tower Case ($80.98 @ Newegg)
Power Supply: SeaSonic G 550W 80+ Gold Certified Semi-Modular ATX Power Supply ($59.99 @ Newegg)
Case Fan: Corsair SP120 RGB 3-Pack w/Controller 52.0 CFM 120mm Fans ($62.99 @ Jet)
Case Fan: Corsair SP120 RGB 3-Pack w/Controller 52.0 CFM 120mm Fans ($62.99 @ Jet)
Total: $1719.86
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-03-03 18:29 EST-0500

Changes made
CPU: i7-6700K -> R7 1700X
CPU cooler: H100i v2 -> NH-D15S
MoBo: Z170A -> X370
RAM: Dominator -> Vengeance
SSD: Corsair Force LS (480GB) -> Samsung 850 Evo (500GB)
HDD: Seagate (3TB) -> Hitachi (4TB)
GPU: Gaming X -> Armor
PSU: Corsair RM750x -> Seasonic G-550
Case fans: Corsair SP/AF series -> Corsair SP RGB series

Reasons why
CPU: Since the AMD Ryzen CPUs got released a few days ago, the first benchmarks are in. For multi-core performance, AMD will beat Intel by a mile. (Mostly due to the high core and thread count.) For video rendering and streaming, the multi-core performance is needed the most, hence why the AMD is logical choice here.
i5-6700K vs R7 1700X: http://cpu.userbenchmark.com/Compare/AMD-Ryzen-7-1700X-vs-Intel-Core-i7-6700K/3915vs3502

CPU cooler: H100i doesn't support AM4 socket, so i had to switch it out. Replaced it with great performing air cooler. If you want to go with AIO nevertheless (even though there's no cooling performance gained with AIO), you can go with Cooler Master MasterLiquid Pro 280 that also supports AM4 socket.
pcpp: https://pcpartpicker.com/product/n7H48d/cooler-master-masterliquid-pro-280-642-cfm-liquid-cpu-cooler-mly-d28m-a22mb-r1

MoBo: Since R7 1700X uses AM4 socket and not LGA1151, had to change out MoBo too.

RAM: Picked a smaller profile RAM that doesn't interfere with the Noctua NH-D15S.

SSD: Replaced your SSD with far better performing one while gaining 20GB of space.
comparison: http://ssd.userbenchmark.com/Compare/Samsung-850-Evo-500GB-vs-Corsair-Force-LE-480GB/3477vsm41604

HDD: Same here, replaced unreliable Seagate drive with Hitachi (HGST) drive while also gaining 1TB.
Further reading: https://www.backblaze.com/blog/hard-drive-reliability-stats-q1-2016/

GPU: Since MoBo and RAM share the black & white theme, replaced GPU with Armor series to math the coloring.

PSU: 750W PSU is way too much for your build. So, put in good quality unit from Seasonic.

Case fans: Corsair AF series fans have terrible performance (both in air flow and static pressure). So, switched them out. Saw that you wanted some eyecandy with your LED fans, so put in SP RGB fans that come with controller. This way, you're not stuck with single color and you can switch your color of fans whenever needed.
Video review: https://www.youtube.com/watch?v=v-ayn9MIs4k

Sadly, Corsair only has 120mm fans in their SP/HD RGB series. So, put in 6x 120mm SP RGB fans that you can mount as follows: 3x 120mm at front, 2x 120mm at top, 1x 120mm at rear.
With 2x 3-pack fans, you'll get 2 fan controllers. You can sell the extra fan controller or keep it as a backup.

Since i'm not in the streaming business, i can't help you out by saying which stream programs are the best.

edit. looked at the wrong thing

I made this with budget in mind.

PCPartPicker part list / Price breakdown by merchant

CPU: AMD RYZEN 7 1700 3.0GHz 8-Core Processor ($328.99 @ SuperBiiz)
CPU Cooler: Noctua NH-D15S 82.5 CFM CPU Cooler ($79.99 @ Newegg)
Motherboard: Asus PRIME X370-PRO ATX AM4 Motherboard ($170.00 @ Amazon)
Memory: Corsair Vengeance LPX 16GB (2 x 8GB) DDR4-2666 Memory ($99.99 @ Newegg)
Storage: Samsung 850 EVO-Series 250GB 2.5" Solid State Drive ($93.99 @ Amazon)
Storage: Seagate Barracuda 3TB 3.5" 7200RPM Internal Hard Drive ($89.69 @ OutletPC)
Video Card: EVGA GeForce GTX 1070 8GB SC GAMING ACX 3.0 Black Edition Video Card ($379.99 @ Jet)
Case: Corsair 200R ATX Mid Tower Case ($54.99 @ Newegg)
Power Supply: EVGA 750W 80+ Gold Certified Semi-Modular ATX Power Supply ($87.99 @ SuperBiiz)
Total: $1385.62
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-03-03 19:07 EST-0500

-1070 for better gaming performance. I know you're just going to game on 1080p, but some games still require more than a 1060.
-Ryzen 1700, because I don't think XFR is worth $70. This is just as overclockable as that one.
-250GB SSD, because I think it's enough, record/render to this and then move it to the HDD for storage.
-3TB HDD, because you're gonna need it, since you're recording videos.
-16GB RAM, I think it's enough, most games won't even use half of it, but because you're streaming and video editing, you're gonna need it.
-EVGA GQ, because it's a good PSU and it's also not expensive.
-D15S, because not a lot of coolers fit AM4 right now, it's the best of the bunch, considering it's price.