Oh, and regarding notifications specifically (not all the other things that need to happen before the app even gets to that -- I mean checking mail and doing it properly) -- two things to check:
- System settings -> apps management -> click into AquaMail -> make sure "Show notifications" is enabled.
This is a system level "kill switch", it can only be changed by the user, but it appears to also change on its own (occasionally, according to some reports here).
- Android 5.0+ has this concept of "priority" and "none" notification "modes". It basically a system-wide "do not disturb" setting.