I reply to this question even it is over one year old since I had the same problems recently. I described how I solved it in German in this article:ios 7 E-Mail-Client mit T-Online. In short the trick is to set an extra password for external email clients (like Outlook or the Apple email client) in the menue of the email center ("E-Mail-Center"), which has its own password - the same as that for the customer center ("Kundencenter").
Set the SSL-decryption setting to "on". Use the following ports:
inbox server: secureimap.t-online.de, Port: 993, safety: SSL/TLS,
outbox server: securesmtp.t-online.de, Port: 587, safety: SSL/TLS.
These settings should be found automatically during the setup of the email account.
And I recommend to set the storage time for emails in the email center from 90 days (default) to "never delete".