A) Quantity over quality. Streaming can use as many cores as you throw at it so 8 cores at a lower clockrate will still run circles around 4 cores because you're never going to get those to twice the clockrate. It's the opposite for games where more cores rarely do much (at least beyond a certain point) and even a minor increase in clockrate gets you more than multiple additional cores.
B) IPC = work done per cycle. So basically speed = IPC * clockrate. Bottom line is if you're streaming the 8700K can't get to the single core turbo of 4.7, it's only going to run at 4.3, maybe 4.4. So best case is something like 15% more fps while streaming and the stutter gone and maybe 20% more while you're not streaming. Unless you overclock.
C) Depends on how much is left of the budget.
PUBG is basically a pile of shit. I mean any somewhat decently clocked quad core gets you 100 fps on very low. No matter how much single threaded performance and cores you add you only end up getting another 20 fps at best and looks like it's GPU limited. On the other hand cranking all settings up to the max only costs 20 fps so it doesn't seem to be GPU limited after all.
RAM is very expensive right now but it might not get cheaper anytime soon. Per GB 16 should be a bit cheaper so you might as well.
To summarize:
TF2: Streaming: more cores will definitely help.
Not streaming: 20% at best if you don't overclock. More if you do.
Modern games: You're not CPU limited even now, so only the GPU will matter.
PUBG: GPU upgrade should help, CPU won't do much. 4790 would be somewhere between the 8100 and 7700K, so once the GPU isn't the limiting factor anymore a better CPU will get you maybe 5 fps (see benchmark).
Witcher 3: Same as modern games. GPU only.
BL2/TPS: Definitely not GPU limited. Should be around the same as TF2 when not streaming. Unsure if more cores will help the minimum fps (averages are already high enough) so I wouldn't bet on getting more.
Basically other than streaming you will get somewhere between 0 and 20% more fps* if you're not overclocking. Considering you will have to buy a cooler and Z mobo (and decided on the 8700K, so unlocked CPU as well) you'd be spending over half your budget (+ RAM) on overclockable hardware. For "up to 20%" this really does not seem to be worth it.
If you only care about the streaming there's cheaper options.
If you want extra performance then actually commit to it and overclock.
*not counting borderlands because minimum fps are different and difficult to predict. Could be more, could be less.