Well it can hold Gaming + Streaming at the same time, it's just you'll probably be at 720p60fps.
And yes, if u have a separate PC, that i3 with 4Gb and a GT1030 will actually stream it without a problem.
But honesty I would suggest going for a GTX instead of an AMD RX. They're a bit pricier but there's a reason for that. The GTX 1070 is the equivalent of the RX580 and also comes with 8Gb of VRAM instead of 4.
Also, I wouldn't suggest NOT going for the R5 2200G, you already have a graphics card so you don't need the integrated graphics, I would rather recommend you go for a Ryzen 5 2600X instead, with which you will be able to Game + Stream at 1080p60fps at the same time.
You chose a pretty good X470 motherboard but it's still...