Thanks guys!
Overclocking does sound good, but I've never really done it myself. The price difference is reasonable though, so I might as well go with 6600K as you suggest. I'm not an expert, so I don't know if all the other details are overly important (when comparing other 6600K's). I've picked this one:
Intel Core i5 6600K Quad Core CPU Processor (Socket 1151, 3.50 GHz, 6 MB, Skylake, 91 W, Integrated Graphics)
So without overclocking my CPU I can live without a cooler? Then I'll wait with that untill I decide to overclock. Compared to my currect computer, this upgrade will be something in the lines of mind blowing, so I doubt I'll complain. Don't have the exact specs with me here, but the AMD Athlon 2x is the main issue.
The reason why I'm going for 2x8 RAM is for future upgrades. I know that only 8 GB is needed for gaming, but I've also heard that Star Citizen likely will need around 16 GB. My friend bought 4x4, and if he wants to upgrade he needs to change all of them. I can simply buy the same sticks of 2x8 in the future.
@jcastro_805, you're recommending 390 which I asume is AMD? From experience and working in customer support, I preffer Nvidia for ease of use. But perhaps AMD has the same features, just more for "advanced" users? Basically it's much easier to troubleshoot issues on Nvidia than AMD.
I'll post my specific case and PSU when I get home. I'm pretty sure it was a 1200 Watt PSU.
Thanks again guys