I want to add an extra field for user age. If age is below 13, it should show another text input for getting mail address of user's parent or other legal guardian. Accounts with age under 13 should only be activated after email verification by said parent.
How can I add this field, validate age and process validation mail?
Simply go to Home > Administration > Configuration > People > Account settings and add fields you need. Use Conditional Fields module to conditionally show a text box.
Now you need to use Rules. See registration rules post for example - what you want is to automatically enable account if checkbox is empty, and send email if checkbox is checked. How to do the very last thing, and how to process verification link, is a material for a separate question.
Questions that may help with further development a bit:
