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Disappeared from the Inbox online via web or from the Inbox on your iPhone?
How long has the issue lasted? If it was minutes ago, please disconnect your iPhone immediately.
Use your computer to access AOL via web (not using AOL desktop) to verify if the emails have been removed from Inbox then the Trashed folder...
It seems like you accidentally deleted something by a mistake then the changes have been synced to server side then other desktop sides.
Good to know that. Just double check if the emails are back on iPhone as well?
I guess you got charged because an expert (me here) have picked up your question from the pool and have been working with you on the issue. However, I am not a billing person, you need to speak with JustAnswer Customer Care for details.
Again, that was not charged by me. If you are not happy with the service, please speak with JA Customer Care to get refund.