They're two separate computers, and the only interface between them would be a video output from the gaming rig going to a capture card on the streaming rig. I don't even know in what way they could be incompatible. But if you require a yes/no answer, then yes, they will be compatible.
Regarding taking the GPU out of the Ryzen rig and putting it into the Intel rig: Ryzen has no integrated graphics. If you take out your gaming GPU, you'll need to replace it (even with a really cheap, low end GPU) in order to get a video output.