If your current motherboard is a socket 1150 then no, it won't work with the 6700. 6th gen intel uses socket 1151 which is completely different. The highest performance cpu for the 1150 socket is the 4790k. Not sure if it will be worth it for you or not to upgrade, the 4790s will turbo boost up to 4ghz and the 4790k will turbo boost up to 4.4ghz. Not a huge speed difference. The 4790k can be overclocked if using a decent aftermarket cooler and a z97 motherboard.
I think you'll be fine, usually as the resolution increases the gpu becomes more important and has more impact. Resolutions around 768p and 1080p there are bigger differences between cpu performance, those differences become less as resolution increases to 1440p and 4k. Going to skylake for the 6700k would mean a new cpu, lga1151 motherboard, an aftermarket cooler if you don't already have one (the k series doesn't ship with a cooler at all) and ddr4 ram.