Mail App keeps popping up in the background in Mac OS Mojave
I have the same issue: 
The google calendar stuff is nonsense. The notification settings suggestion doesn't work.

I've narrowed it down to the following:

I normally hit the red cross button on the viewer window once I have finished looking at my mail, this closes open viewer windows but leaves the mail app active in the dock:

So if I have mail running, but no viewer windows open, then I get a viewer window randomly popping up to the foreground a few times an hour. 

If instead of hitting the red cross button, I minimise mail using the minus button, then I don't get any viewer windows randomly opening. In this scenario, the mail viewer window is minimised either on the right hand side of the dock or into the application icon itself, depending on your settings (System Preferences > Dock > "Minimize windows into application icon").

So I have changed my habits to work around this annoying bug...