You should be fine with stock cooler if you are not overclocking but stock coolers are louder.. And Yes you need to have cpu cooler. either stock or aftermarket
Inside your cpu box there should be an included stock cooler. If it doesn't come with a cooler (tray), then buy an aftermarket cpu cooler. For around $20 you can get a cpu cooler that cools a lot better and quieter than the stock cooler.