Acer Predator G3 710 Compatibility Question

No the Intel and AMD processors require different motherboards. Which i5 do you have in yours? You likely can upgrade to an i7, but there will only be a couple of i7 models you can use.

Number of cores is only one factor to use. If your current i5 is a seventh gen (as in i5-7xxx) then you are correct as to which processor is tops.
The i5 has four cores, but only can handle four instructions at once, the i7 can do eight at once with the same four cores. The i5 has 6MB of layer 3 cache, the i7 8MB. That means the i7 is actually a lot faster in relation to the i5 than just the number of core and base clock rate would show. The other big factor is the GPU. Which graphics card do you have? A lot of the graphics work in streaming is off loaded to the graphics card and isn't affected all that much by the CPU.

So your processor is a sixth gen i5, not a seventh. You can't upgrade to the i7-7700K without a motherboard swap since you need the newer chipset for the 7th gen processors. That means an i7-6700K is the fastest you can use, not the i7-7700K. How much memory do you have? Are you booting from an SSD or HDD? Your GPU is certainly fast enough to stream without glitches unless you're going for really high resolution.