Registration form customization to meet COPPA and similar "under 13" laws?
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:


How do I Send HTML Emails via Rules?
how to create rules to send email notification for node reference content?
and more...