Thank you for the log.
First, your mobile operator is too happy to disconnect the app's network connections. Sometimes they do that (an "optimization").
Since you're using push mail, the app notes those errors, and retries, but only a limited number of times.
I would recommend restricting push mail to WiFi only (and so the app will do scheduled checks) in app settings under "mail, receiving", however:
Second, the app is also failing on initial connects to all your mail servers -- and every time it happens during SSL negotiation (not initial socket connect).
Don't know why there is any difference vs. when app is in foreground -- but this makes me think of Android's "restrict background data" (in system settings under Data usage) or perhaps a battery saving app (third party or built-in).
And to reiterate -- the app is trying to do what it's supposed to do, but its network connections are disconnecting or not establishing at all.