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Do you share the Apple ID with your wife? Make sure that Mail is disabled on her iPad in Settings - Mail - Accounts, in your email account.
For the email account that can't move your emails to trash, here a few things to do. Try removing an email after every step, maybe one of them fixes the problem, without the need to try all of them:
Turn off the account in Settings - Mail, wait for a few minutes, then turn it on again.
Remove the email account from your iPad (delete it completely), then set it up again.
In Settings - Mail - Accounts - (the email account with the problem), go to Advanced - Mailbox Behaviours - Deleted Mailbox, and make sure there's Trash visible. And in Move Discarded Messages into, select Deleted Mailbox.
Hope that helps.