Do the lag spikes/FPS drops only happen in online gaming? If so, it could be the Internet connection (either on your end or on the server's end) that causes those lags/drops.
Also, as pointed out by the others, you have a very powerful system. And you're not going to see any real performance upgrades by switching to a core i7 chip, or even moving from your Skylake chip to a Kaby Lake chip:
-- Battlefield 4 (http://www.techspot.com/review/734-battlefield-4-benchmarks/page6.html): the game is over 3 years old now, but back then there was not only little to no performance differences between the core i5 & core i7 chips, there were little to no performance differences between the Ivy Bridge & Haswell chips, & that was with the chips running from 3.3 to 3.6GHz. Not to mention that an Ivy Bridge core i3 (i3-3220) was able to hang with those chips. Basically, just about any 2C/4T or better CPU with over 3GHz per core will max out that game.
-- Battlefield 1 (http://www.techspot.com/review/1267-battlefield-1-benchmarks/page4.html): obviously a much newer game, & their CPU testing was limited due to EA Origin's "hardware change" issues. However, when the "better" core i7 (with a faster clock speed to boot) puts in the same performance as your core i5 (with the only difference being better minimum FPS on DX12), you're not going to see improvement either.
-- For Honor (http://www.techspot.com/review/1333-for-honor-benchmarks/page3.html): only 1 of the 3 that there were "measurable" improvement in your FPS by going from the i5-6600K to an i7-6700K. However...the problem is that the performance increase was very, very small (~5% for minimum FPS, ~3% for average FPS), & at 1080p your performance is dramatically held back by your GTX 1060 anyway (http://www.techspot.com/review/1333-for-honor-benchmarks/).
Can't speak to CS:GO's performance, but even though it's supposedly a CPU-bound game I'm betting that your GTX 1060 would hold you back from super high FPS anyway. So what it comes back to is either a) these spikes/FPS drops are due to the connection to the server (which you can maybe work on if it's on your end, but not much you can do if it's on your ISP's end or on the server's end), or b) the spikes/drops are happening because your CPU and/or GPU is getting thermally throttled (in which case you should monitor their temperatures while gaming & idling to see if any temperature spikes match up with the lag spikes/FPS drops).
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