Here's a build I put together that you might want to consider:
https://pcpartpicker.com/list/kd4Rkd
-If you're planning on streaming, Ryzen is the way to go; don't bother with the X series since even if you aren't planning on overclocking them, the extra cost isn't worth it (plus overclocking is fairly easy if you do find out later you need some extra power from the CPU)
-The stock cooler that comes with the Ryzen is fine for non-OC cooling; given that your budget left little room for the build due to the additional monitor and other accessories, this is something that can be changed at a later date should you change your mind about overclocking
-While the stock cooler is fine, you'll want to use a quality thermal paste since the stock paste isn't great
-4TB will be enough to hold your games and video footage; you could opt for a RAID array with 2 2TB drives for better performance or for 4 1TB drives in RAID 5 but given your lack of interest in overclocking, i'm assuming you don't want to go through the task of setting up a RAID 0 or 5/6 array
-The HDD is designed for NAS storage but will work just fine in a regular PC
-The MX Master is a fantastic mouse; I personally use one and recommend it for both gaming and production i.e. video editing
-16GB is more than enough for gaming (for now), decent for more intensive activities like video editing; ideally you'd want the maximum amount your motherboard will allow (64GB) but you can always add in more when prices drop
-You didn't specify what resolution you wanted to play in so I assume you'll be fine at 1080p; the RX 560 is a decent card for that resolution but if you find a 1060 on sale (or even an older GTX 770 or 970), that would be the best purchase you can go with given how ludicrous GPU prices are at the moment