The reason why I don't have a case with my build is because I already have one. It's the NZXT 410 Phantom.
I'm assuming you mean after market CPU cooler? If so then no, you won't.
You will ALWAYS need a cpu no matter what. The CPU is what makes everything run on the computer. You could say it's the computer's brain. But if you are talking about a after market CPU cooler then I would recommend one such as the Cooler Master Hyper 212 EVO because AMD CPUs heat up very qiuckly.