yeah i would get the i5 6600k. it will give you a nice boost in performance and give you two more cores to work with in games. Quad core is pretty much the new gaming gold standard.
For strictly gaming, there's not much difference. The i5 does a little better in more CPU intensive games, but unless you overclock the i5, I don't think it would be worth it.
For strictly gaming, there's not much difference. The i4 does a little better in more CPU intensive games, but unless you overclock the i5, I don't think it would be worth it.
https://www.youtube.com/watch?v=xJuadijvTSU
I see an issue with the video you posted. with gta V the vram usage is fairly different
(3458mb vs 2408mb) especially at the beginning when no cars are viable, could just be a quirk but makes me wonder. it does get closer later in the game and the others are all close enough to be difference in NPCs on screen/etc. However there is one other thing i see in the video. IMO the video of the i3 6100 is chopper when at similar frame rates. Which is what i would expect with 2 cores with 2 virtual cores compared to 4 full fledged cores. So its a question of what the OP is willing to pay for. I would take a slightly lower frame rate that is smoother over a higher one that doesn't look as smooth. Regardless it is something for the OP to consider.
the stuttering the OP mentions might be cleared up with a quad core like i mentioned seeing above...might not though as well. If it were me i'd get the quad core for longevity/smoother frame delivery but its not an easy call.