No, respondents can't print a record of their answers after they complete a survey. However, if you allow respondents to edit responses, they can go back to review or change their responses.
As the survey creator, you can export and print a PDF of an individual response in the Analyze Results section.
Depending on your situation, you might consider creating a form instead of a survey to collect the data you need. Forms are useful for collecting information and keeping a record for both the form creator and the respondent themselves—whereas a survey is best for collecting feedback and making decisions based on aggregated results. For example, think of an order form—customers want to keep a record of what they ordered, and the company needs to keep a record of the response to follow through with the delivery.
With Wufoo, an online form builder, you can set up confirmation emails so that respondents automatically receive an email with all their answers, and any custom message you'd like to include along with that.
There's no way to automatically generate a certificate of completion for respondents, but there are two alternatives: