It doesn't sound like it is a problem of the app. Rather, it sounds as a combination of some server settings and some other action happening from a different e-mail client.
Basically, Aquamail (and any IMAP mail client) is a "window" into what is on the e-mail server. It shows messages on the server, and if the message is deleted on the server, it disappears in Aquamail. So, in the situation you've described, something deletes the message on the server between the time when Aquamail first downloaded it (and you saw it in passing), and when you look at it again.
So, hence an additional question: do you have any other e-mail client that might be connecting via POP3 (cell phone, tablet, laptop/desktop, webmail, etc.)? If you do, - once it downloads e-mails, they may be deleted on the server. (Either it is configured in the client to delete downloaded messages, or in some cases the POP3 server will delete or hide messages ones downloaded via POP3, even when the e-mail client would be configured otherwise.) This type of scenario has been frequently seen among e-mail users, and it is not related to Aquamail per se, but would be the case with any client.
And it sounds like this mysterious disappearing happens only with the unread messages. This would be consistent, if the server is configured to serve to the POP client only unread messages. I'd expect that that additional POP3 client is configured to check messages very infrequently (or you tend to read your messages on the phone as soon as they arrive), - otherwise you'd be seeing more messages disappearing.
By the way, - do you have some other e-mail client accessing e-mail via EWS (e.g. your laptop's Outlook). I haven't worked much with the Exchange server, but I wouldn't be surprised that there is a scenario where something similar to what I described with POP3 client/server might be happening due to the EWS client/server, and how the interaction between EWS server and IMAP is configured internally.
PS. While I was writing this, Kostya has responded to you. I post my suggestions anyway in case they will help you understanding possible interference effects.