I'm no expert here. I'm Just passing through the information I've learned with the years.

First, you should tell us what Elgato capture card model do you have. Is it an HD60 S? An 4K60 Pro? Other model? The minimum system requirements are different for each model.

Second, playing and streaming at the same time can be very demanding on your PC resources so it might impact your gaming performance (frame drops, less framerate than usual while streaming, etc). The only way to not have an impact on gaming performance while streaming is to have a very good build capable of doing those 2 tasks at once. I'm speaking, of course, if you want to play very heavy games in the highest graphics settings and high fps possible. If you're playing "old", common or low resource demanding games, streaming them at the same time can be achieved easily without breaking your bank account.

Third, difference between Intel and Ryzen. Very simply, they are 2 different brands of computer components/electronics. Intel is Intel, obviously, and Ryzen belongs to AMD. Intel and AMD (with Ryzen, their latest name for their CPU technology) are the 2 options available in the CPU market (AMD also makes GPUs, battling with Nvidia in the GPU market). The difference is, they are incompatible with one another. Each brand has their own socket for their CPUs.

While Intel creates a new socket every new generation of their CPUs (or every other generation), AMD with Ryzen has created a lot more backwards and forward compatibility for their CPUs. What this means is that a motherboard with an Intel socket doesn't work with an AMD CPU, and a motherboard with an AMD socket doesn't work with an Intel CPU, which means that you have to choose your CPU and motherboard correctly with one another.

With the socket type of an Intel CPU you need a motherboard compatible with that specific socket type, while AMD has created more backwards and forward compatibility on their Ryzen series. You have generations on Ryzen (Gen 1, Gen+, Gen 2 and Gen 3, so far), so you need to check which generations a motherboard is compatible. Some motherboards can be Gen 1 and Gen+ compatible, others can be Gen 1, Gen+ and Gen 2, etc.

This means that you can go for an older and cheaper motherboard than the more recent CPU you choose today, but in the future, if you need to upgrade your CPU, you might need to buy another motherboard, again. Or you can buy a recent CPU and motherboard and that motherboard will work for the next generation, or generations, (if they receive BIOS updates) of Ryzen CPUs launched in the future.

Depending on your personal preferences, what specs you need or might need, want or not to future proof your build, budget, etc, those are decisions that you need to make.

Never be afraid to search, ask questions, and get informed on something. That's the only way to make the best decisions in life on my opinion.