No, it is not required to purchase privacy protection, nor even a domain if you are applying for an AdSense account using a blogspot URL.
 
Approvals are  not based on either of those things. They are based on the content of your site – the quality of the content, and the quantity.
 
See the following links for information on the requirements for websites/blogs in terms of contents and quality.
Minimum content requirementsMake sure your site has unique high quality content and a good user experienceWebmaster quality guidelines for thin contentWebmaster quality guidelines
Also these pages for policies and content guidelines:
Eligibility Requirements
Make Sure your Site is Ready
Site Content Issues/Problems
AdSense Program Policies
Google Publisher Policies
Google Publisher Restrictions