Yes on both counts. An email address and a domain name will help to build your credibility. They prove that you are serious about your work and give your clients another way to communicate with you.
I encourage you to get your own personalized domain name and use it for your email address and your webpage. There are many free email services and free web hosting services out there and you may be tempted to save money by choosing one of these. Before doing that, consider these two email addresses:
firstname.lastname@example.org vs. email@example.com
Who would you rather do business with?