Figure it this way. Each of the cores on a Ryzen cpu are pretty much equitable to the Haswell/Devils Canyon Intel cores. So what a R5 1600 represents is basically equitable to a 6core/12thread i7-4770k. For gaming purposes, that's @10% less than what a skylake/kabylake cpu can do with single thread power. But. You are not looking for single thread performance as much as multiple thread performance. And that 4c/8t i7 cannot even compare to 6c/12t. For sheer amount of work capacity, a Ryzen i5 1600 will dust a i7-7700k. If you drop down to a R5 1500, now you are matching an i7 core for core, and Intel faster throughput will make a difference. It's 4 threads. That's the same amount of processing power as an i3 dual core HT cpu. That's a huge difference in ability.
Stick with the 1600, even if you gotta eat Ramen noodle soup for a month, it'll pay off dividends for what you need the pc to do.
Not a fan of Gigabyte boards, when they work right they are great, probably the best there is in each area, but if there's any quirks, you go nuts. Asrock is decent, even (imagine my surprise) the Biostar x370 is very good, but really it's mostly a competition between Asus and MSI, and MSI is winning in the more budget areas with the Gaming pro, Tomahawk etc.
Looking at ddr4 ram, there's 2 considerations. For your needs, more is better, if you could squeeze 32Gb, go for it. Generally 16Gb is the minimum recommended, 8Gb absolute lowest and thats only if you have no option. 2nd consideration with a Ryzen is the cpu. It's new 'Infinity Fabric' version of HT makes much better use of higher speed ram than Intel does. 3200 is about the limit of reasonably priced ram. Over that and prices get ridiculous. Personal favorite is g-skill Trident-Z. So far they have shown the best compatability with Ryzen cpus, along with G-skill ripjaws V. Last I heard, amd was still working on issues with Corsair Vengeance LPX, but that either has changed, or might change soon. So 16Gb of g-skill Z or V is still the best bet.