Does JotForm keep a register of emails and addresses of forms sent to?

I am not sure if I have understood your question correctly. Do you meant to say that you want to track email sent when you reply / forward submissions from form's submission page?

Please note that our email history does capture all the submissions reply/forward. You can check your account email history to see the all the emails sent from your account. The following guide  should help you how to check email history:   https://www.jotform.com/help/293-How-to-View-All-Your-Form-Email-History 

If you have configured email alert in form, you will receive submission email after the form is submitted. You can reply / forward submissions form your own email inbox and you will be able to track all the submissions you have replied.

Hope this helps.

Do get back to us if you meant to ask something else.