I only owned my ASRock Extreme6 for a few days, but I gotta say I really liked the bios layout and overall the board worked great for overclocking. I have a couple other skylake ASRock boards running as well (H110/H170 boards) that run great without issue. I'd not hesitate to buy another.
My current gaming rig is running a Gigabyte Gaming 7, and overclocking support has been good on it. The bios is a little less intuitive than the ASRock, it takes some getting used to. Overall, it's been a good board though. Another one I'd not hesitate to buy again.
Between the two though, I prefer the ASRock bios.
I've used previous versions of the ASRock Extreme4 back when I was running the... 2500k, I think? Good motherboards, would definitely recommend. Right now I have an MSI Krait edition for my 6600k and it works fine, but I agree with your that the ASRock bios is good, I like it better than the MSI.
For the OP, as far as Intel chips go, for straight gaming right now the chip of choice today would be the i5-6600k. The i7-6700k has only two advantages over the 6600k: a higher base clock (a difference you'll make up in overclocking anyway), and hyperthreading. Hyperthreading doesn't help for gaming performance as there just aren't games out there that will utilize >4 threads. However, streaming can be helped by that feature, so if you do a lot of streaming the i7 is a good choice.
Why not the the 2011-3 chips? (i.e. the 5820/5930/5960?) Personally, I think even their theoretical performance just isn't good enough for the price premium. You'll spend 2-4x as much on the chip but get nowhere near that increase in actual performance. The larger core counts are somewhat offset by the lower clock speeds, and in gaming situations the individual core clock speeds are of greater importance than core count. After all, if your game only utilizes two cores, you want two faster cores instead of six slower cores (because four of them will idle anyway). That said, I'm not personally very familiar with the CPU load in encoding video for streaming, so some more expert opinions on that would be useful.
And regarding switching to Intel or not... yes. I used to buy AMD, but Intel has just managed to put out better processors at virtually every price point these days. Other than an extreme budget build, there's just no AMD processor I recommend these days. The biggest dead giveaway on CPU performance comes from AMD themselves. AMD makes GPUs as well as CPUs, and in marketing those GPUs will often release benchmarks. Find the right marketing material and you'll see that
AMD uses Intel processors when benchmarking their GPUs.
