I still have Outlook 2010 and therefor no outlookßcom-account. But everybody can get one.
So I opened an outlook.com-account, added it to iPad, iPhone and Outlook on my Windows-PC. Downloaded Microsoft To Do on my Apple-devices.
To Do syncs nicely with Reminders (!) and Outlook Tasks on PC. So I can work with Reminders as before and put To Do on the last screen on the Apple devices, just because it is necessary.
With the outlook.com-account there was a new Calendar called outlook.com. You can see it in Calendar on the Apple-devices and in Outlook on the PC. Same with Contacts.
So I copied contacts and calendar-entries into this calendar and ... Heureka!
For my part, I am happy again. Everything works like before and no iCloud needed anymore.
The Outlook-app for iOS is good, but I am used to the generic ones from Apple, so I will not switch at the moment. But I am satisfied that there is a really good alternative in case we will need it in the future.
Thanks again. Without your post I had never been encouraged enough to try this changes.