Separate cooling fan for core i5?

If you bought a retail CPU then it will come with a standard heatsink and fan. If you ordered the OEM version you will have to order another.

If you plan on overclocking or just want a better cooler then you should buy a better one, as the stock ones aren't particularly great. For the most effect use a liquid cooler. You can get them for a similar price of an air cooler now, and the performance is much higher than air due to liquid having a higher thermal capacity and better thermal conduction properties than air.

No but i recommend doing so,the stock cooler is rubbish.Get a Hyper 212 cooler,it have thermal paste bundled in a syringe.101 is also ok but the 212 will be the best.

The box should come with one unless your order invoice says "OEM".