For a much lower price you can get a Ryzen 5 3600 and compatible motherboard, which would perform better.
You could also save money by using the Ryzen stock cooler if you do this
I would also recommend getting a larger storage device, because 250 GB is not going to last very long.
A 500 GB (or 1TB if you want) SATA 2.5 inch SSD (a Samsung 860 Evo for example) would probably be about the same in terms of performance and have a lower price. Also, using an additional HDD for files you don't need to load quickly would save space.
If you do have a budget, I really wouldn't recommend spending $130 on a DVD drive.
The DDR4 2133 RAM you have is really overpriced. I would replace that with Corsair Vengeance LPX DDR4-3000, which performs better and has a much lower price.
The case you have is large and expensive, I would say a Mid Tower ATX case would serve the purpose well and save 60-100 dollars. For example the Phanteks Enthoo Pro M TG would cost 90-100 dollars and still have a 5.25 inch bay and space for what you need.
I don't know much about AMD cards but if you do get an Nvidia card get an ASUS, Gigabyte, or EVGA card
A GTX 1060 6 GB would be adequate for most 1080p60 gaming, and if you want 1440p, you can spend more money and get an RTX 2070.
Finally, the power supply:
I really don't recommend cheaping out on a power supply because if it blows, it likes to take a couple other components with it. I would choose a Corsair or EVGA 550W or more 80+ Gold (or more) rated power supply.
As for your planned budget of $1600, that could be put to use far more efficiently.
Here's a ~$1600 list with a Ryzen 7 3700X and RTX 2070 for example
https://pcpartpicker.com/list/vYJWV6
Edit: I may be mistaken, but if you already have a windows device, you can download windows 10 ISO files for free to install windows on a fresh device (it just wont be activated, which would have some cosmetic limitations)