1. Does the CPU come with a CPU cooler? and if not what kind of CPU cooler should I opt out for.
Cpu usually comes with heat sink and fan, it's optional to buy Cpu Coolers.
2. Is the selected motherboard worth it or is there a cheaper option available as I am slightly above my budget.
That motherboard is good, there are cheaper options but that also causes cheaper performance, if u want the best of the best, a good mother board should do.
3. Is 1866MHz RAM required for gaming these days or should I be looking at 1333MHz or 1600MHz as a cheaper option.
If you're looking for better gaming play and love new games with fantastic graphics then you're good
4. I have no clue if I should get a full tower, mid tower or mini case. I will only be using my computer at home with no plans on taking it to LANs and such. Do you guys have any case recommendations?
I personally think full tower are too big, I would prefer mid tower cases so I could bring it to places.
5. Is the power supply sufficient? On the system build page it says I require ~350W. I thought keeping the 600W PSU will allow me to upgrade if I feel like it in the future.
600W is good, that way you don't ever have to go back and get a new one.
6. Is upgrading to a i5 6600k skylake CPU worth it? I looked at some benchmarks and there is around a 10FPS increase on AAA titles for only a $15-20 increase in price. It also allows for a higher RAM clock speed.
The better the cpu, the better gaming. You'll definitely know the difference once you try it.