Is it Appropriate to Include Personal Correspondence in the Appendix of a Thesis?
You can only publish email correspondence in your thesis if all people involved in the email communication agree.  Anything else is highly unprofessional and also unethical.
On the other hand, if you have clearly demonstrated that the manuscript had an error, there is no need to add those email.  You could write that this has been confirmed by the author and cite a &quot;private communication&quot;.
Regarding the emails, there should be no problem to save the emails even if your account is deleted.