1. A domain is not necessary, it is optional. It will be easier and cheaper if you use a URL shortener instead.
2. It does not need to be private but that is up to you.
3. Once you buy the domain you simply need to set up a redirect (you can ask godaddy support how to do this if you are unsure).
4/5. You will be using your CPA offer link as the redirect link on your domain and then use the domain link on your flyer. This means that people will go to your domain and then will be sent directly to the CPA offer.