Thank you for providing all the great details, SteveBMe! It would definitely be inconvenient to have deleted emails repopulate. Don't worry we will do our best to resolve this concern.
I know you had mentioned that there was a Verizon update last year. Can you please clarify as far as what type of update was done (software/app)? Have you attempted to sign out of all email accounts and sign back in? When you delete an email on your phone and it repopulates, have you been able to confirm if it shows back up in the inbox while logged into your email on a computer? Are you able to test this both ways, deleting an email from the computer and seeing if it still shows up in your phone's email?
Please let us know the results of these steps. We definitely want to get this fixed.
Follow us on TWITTER @VZWSupport
If my response answered your question please click the _Correct Answer_ button under my response. This ensures others can benefit from our conversation. Thanks in advance for your help with this!!