The log seems to show that emails are being sent, it looks like it's finding the servers to deliver mail to - but they are not getting there.
I started with exim4, but my other server had postfix, so I tried that, the results are similar. In exim, the emails were frozen. Basically not returned undeliverable, but not delivered.
Here is the log when I tried to send 3 emails with a php script - one to a hotmail account, one to gmail account, and another to a company account that's hosted on gmail.
Aug 2 21:59:01 myawesomedomain postfix/pickup[28232]: 9B4C79DEDB: uid=33 from=<www-data>
Aug 2 21:59:01 myawesomedomain postfix/cleanup[28272]: 9B4C79DEDB: message-id=<20160802215901.9B4C79DEDB@myawesomedomain.com>
Aug 2 21:59:01 myawesomedomain postfix/qmgr[27728]: 9B4C79DEDB: from=<www-data@myawesomedomain.com>, size=380, nrcpt=1 (queue active)
Aug 2 21:59:01 myawesomedomain postfix/pickup[28232]: 9E9849DEE7: uid=33 from=<www-data>
Aug 2 21:59:01 myawesomedomain postfix/cleanup[28272]: 9E9849DEE7: message-id=<20160802215901.9E9849DEE7@myawesomedomain.com>
Aug 2 21:59:01 myawesomedomain postfix/qmgr[27728]: 9E9849DEE7: from=<www-data@myawesomedomain.com>, size=378, nrcpt=1 (queue active)
Aug 2 21:59:01 myawesomedomain postfix/pickup[28232]: A1E649DEE8: uid=33 from=<www-data>
Aug 2 21:59:01 myawesomedomain postfix/cleanup[28272]: A1E649DEE8: message-id=<20160802215901.A1E649DEE8@myawesomedomain.com>
Aug 2 21:59:01 myawesomedomain postfix/qmgr[27728]: A1E649DEE8: from=<www-data@myawesomedomain.com>, size=383, nrcpt=1 (queue active)
Aug 2 21:59:01 myawesomedomain postfix/smtp[28278]: connect to gmail-smtp-in.l.google.com[2607:f8b0:400d:c06::1a]:25: Network is unreachable
Aug 2 21:59:01 myawesomedomain postfix/smtp[28282]: connect to aspmx.l.google.com[2607:f8b0:400d:c00::1a]:25: Network is unreachable
Aug 2 21:59:02 myawesomedomain postfix/smtp[28278]: 9E9849DEE7: to=<myexcellentemail@gmail.com>, relay=gmail-smtp-in.l.google.com[209.85.232.27]:25, delay=0.57, delays=0.01/0.01/0.29/0.27, dsn=2.0.0, status=sent (250 2.0.0 OK 1470175142 u184si197895qkh.319 - gsmtp)
Aug 2 21:59:02 myawesomedomain postfix/qmgr[27728]: 9E9849DEE7: removed
Aug 2 21:59:02 myawesomedomain postfix/smtp[28277]: 9B4C79DEDB: to=<myexcellentemail@hotmail.com>, relay=mx1.hotmail.com[65.55.37.104]:25, delay=0.65, delays=0.01/0/0.24/0.39, dsn=2.0.0, status=sent (250 <20160802215901.9B4C79DEDB@myawesomedomain.com> Queued mail for d$
Aug 2 21:59:02 myawesomedomain postfix/qmgr[27728]: 9B4C79DEDB: removed
Aug 2 21:59:02 myawesomedomain postfix/smtp[28282]: A1E649DEE8: to=<companyemailhostedatgmail@mycompanydomain.com>, relay=aspmx.l.google.com[173.194.207.27]:25, delay=0.89, delays=0/0/0.49/0.39, dsn=2.0.0, status=sent (250 2.0.0 OK 1470175142 s37si2836049qta.36 - gsmtp)
Aug 2 21:59:02 myawesomedomain postfix/qmgr[27728]: A1E649DEE8: removed
I found them in the spam folder - listed as being sent DAYS after they actually were sent, and unflagged one, and now they're getting where they should be going. I'm lucky they're admin emails, and only have to go to a fixed number of addresses, I'll make sure the users know that they could end up in spam.
– user1632874 Aug 03 '16 at 14:18