Trash and unread mail boxes show 700 non-visible emails Can't get the boxes to empty or the number to change or disappear. Don't know if there are any real emails stored somewhere or not
This is easy to sort.

Put the iPad in airplane mode and then power it down.

Then power it up and don't connect to any networks.

Open mail.app and see what the status it. It will warn you, perhaps several times with different wording, that you are offline. Remain offline and then get a read count. You should be able to mark messages as read and then decide when to connect to a network - hopefully a fast one to see that things then sync up.

As an alternative, power off the iPad and use webmail to get a feel that the mail service is working, then work on the iPad as above.