IOS 8.1.3 – Return to Inbox After Deleting

My guess is that one of your clients is removing emails from the server - last time I used POP this was the default, and it was one of the best reasons for switching to IMAP. If any of your clients is doing this then those emails will not be appearing on your other clients. Have you checked this on each client that you access those servers with?

I know you didn't say you access those emails from multiple places but it's a common thing to do.

iOS Mail app doesn't function like you are hoping it will unfortunately. The best suggestion I have is to add this mail account twice in iOS Settings, once using each email address you prefer it to have.