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 "private communication".

Regarding the emails, there should be no problem to save the emails even if your account is deleted.

In some cases you can also write:

The mistake described above has been confirmed [15] by the authors of [12].

where the extra citation may read something like:

[15] Shot, Big and Fry, Small. Personal communication, September 9th, 2020.

A mathematical argument speaks for itself and its validity does not hinge on the blessing or confirmation of a single individual, even if the argument corrects a mistake that individual has made previously. So I think your premise that you need to include anything about the error being confirmed by the original author as some kind of “supporting evidence” is simply false. The fact that the author confirmed what your said is something that can be mentioned if you’re doing it as a way to save the author a bit of embarrassment, but is otherwise irrelevant. If you simply provide an explanation of the error and the details of how it is corrected, that will completely suffice to convince any readers of the validity of what you are saying. No emails need to be quoted.

As for your literal question: no, as a general rule it’s a terrible idea and quite inappropriate to quote a personal email in a thesis or other publication without explicit approval from the author of the email. But as I said that’s sort of the wrong question to ask for your particular situation.