Help me start my build

1b. Ryzen 3 1200 will do, but it won't last as long as the Ryzen 5 1600 will. If you don't mind upgrading the CPU in the future, then the Ryzen 3 1200 is a great option as well. Avoid the 1300X, it's the same chip.
2a. That's why I recommend B350. They're cheap, and if you aren't doing multi-GPU, they have all you need. Any modern motherboard will most likely support a new GPU 2 years from now. Get the B350 if you don't like multi-GPU(which no one should like, IMO).
2b. Yes it would.
3a. 2x8. Better for higher clock speed. 4x4 really stresses the memory controller on Ryzen, wouldn't run at high frequency. Get 2x8 sticks, it's the best configuration for Ryzen today.
3b. Depends on you, but apparently G. Skill runs better with Ryzen according to some. Some would disagree with this, so really it depends on the specific kit you get. But try to get G. Skill if you can, otherwise any would work just fine.
4a. In the future it might. 3 GB is less, but the 1060 is still faster in most games, barring VRAM-heavy games like Resident Evil. 1060 3 GB really struggles in that game, but in others, the two cards are neck-and-neck. For long-term, get the 4 GB RX 480, but if you're upgrading soon then the 1060 3 GB is better.
4b. DX12 still isn't fully here, but yes it would. If you really prefer it you can get it. Without having both cards side-by-side, you won't ever be able to tell the difference between the 1060 3 GB and RX 480 4 GB, they're that close. Get the one you like, you'll be happy with either.
4c. Forgot to tell you, the RX 480 is out of production now. If you're still getting one new, bear in mind it's probably a little old. The RX 580 is the newer equivalent, and doesn't cost more. But right now they're out of stock everywhere. Either way, all three cards(1060, 580 and 480) will perform good.
5. If you like the extra VRAM and future-proofing of the RX 480, get that. GTX 1060 3 GB mini is also good, but you seem to dislike the lower VRAM. Take your pick, both cards will serve you well for 1080p.
7a. It might be, it might not be. Can't say right now, the launch is still far off. But Intel will want to compete with Ryzen, so if they price everything higher they won't sell all that well.
7b. Again, maybe, maybe not. Ryzen is still pretty cheap right now, so I wouldn't wait for price drops except maybe for Black Friday sales. I doubt Ryzen's going any cheaper, considering how well they're selling at current prices.

About the GPU confusion, I actually think you should get the RX 480, despite what I recommended previously. You will probably keep the GPU for quite a while before upgrading, and the better DX12 support and extra VRAM will help the GPU last longer. Get the dual fan RX 480, and try to overclock just a little bit.

1. Honestly, any case will do. Both the components will come with very good stock coolers that will keep them cool, provided there is enough airflow in the case. You can increase airflow in any case with case fans. So, get the case that you like, and just add case fans to increase airflow. Here's my recommendation:

https://pcpartpicker.com/product/ms6BD3/nzxt-case-cas340wb1

And these case fans:

https://pcpartpicker.com/product/wL3RsY/cooler-master-case-fan-r4l2r20arr1

Add as many as you like, do keep in mind that the case comes with two included, and supports two additional 120's in the front. You can check the full case fan support here:

https://www.nzxt.com/products/s340-black

2. The same case will allow for overclocking just fine. Overclocking increases temps, but not enough to justify buying another case - adding better/more case fans to the current one usually reduces the temps.

3. For a B350 board, go with either the Asus B350-Plus or the MSI B350 Tomahawk. Both are great boards, the Tomahawk is usually a bit more expensive, but also has better features. Both will overclock just fine, and shouldn't give you problems unless they're DOA(DOA products are unavoidable, can't do anything to ensure you don't get DOA products).

4. Corsair memory will also run fine, but get at least DDR4-3000, because in games higher clock speeds help Ryzen perform a little bit better. You can usually run 3000 MHz RAM at 2933 MHz on any good board.

Honestly, there isn't much I can do to help with the motherboard - it's totally a personal choice. If you think gigabyte boards might not be as long-lasting, get an Asus - it really doesn't matter as much as you may think. Also, the B350-F is a little better than the Gaming 3 - the Gaming 3 competes with the Asus Prime B350-Plus. If you can afford the B350F, get that since it's a better board with more bling and features.

Yes, the M stands for Micro-ATX.

Get either of the 1060 6 GB cards, they're better than the RX 570 performance-wise. Out of the two 1060's, pick the one you like, the performance will be quite similar - pick the brand you trust, the brand which has better warranty, or simply the card that looks prettier to you - the performance is the same for all practical purposes. The top priority for picking between the two should be price, but it really depends on what you think.