More cores are rarely gonna be more useful in gaming, but that's not as true as it used to be. Some games take advantage of it well--it's just that a ton of console ports (which make up the overwhelming majority of system-choking AAA titles these days) tend not to multi-core very well. So, one of the chief selling points of an i7 is sorta undersold by that. Now, if you plan to do stuff like video encoding or, like you said, streaming-while-gaming, it becomes a lot more attractive. If cost's not an issue and you think that'll be a significant use case for you, I say go for it!
The Hyper 212 EVO is a perfectly respectable cooler and will let you achieve good overclocks with an Intel K processor. I think that there are newer options that are better, like the Cryorig H7, which I bought this year, costs almost the same as the current-gen Hyper 212X from CM, and is generally considered a better performer and marginally quieter to boot. But if you've got a 212 EVO already? Stick with it; water-cooling's for folks interested in cool-looking gear, edge-case overclocking levels, or e-peen-measuring contests ;)
Overclocking's not super necessary, but it can help an older processor eke out another year or so of useful life. If you're gonna be pushing the budget picking up an i7, for instance, and really want your money's worth, it might be a useful step-up. A modest-but-noticeable overclock's almost certainly not gonna cause any kinda issues that'd crop up during its lifespan. And yes, the auto-overclock settings on modern mobos are pretty good, and if you don't want to pull out every last ounce of performance you possibly can, will probably be good enough. Almost certainly. Unless they aren't for you and you've gotta go muck with manual crap after all :)
That said, realistically, outside of a handful of super-CPU-bound games (e.g., Cities: Sklylines, some of the recent Total War games, etc.), an overclock's not gonna mean much more than a couple FPS here or there. If the money you save on that let you spend on a better GPU 6 months from now, that'd probably be a better buy, overall.