I'm using Android 6.0.1 and Outlook for Android 2.1.33. My data is on MS servers. After several experiments I got my phone normally syncing contacts, calendar and e-mails between outlook.com (= desktop computers with Office365), phone's built in calendar
and contacts and Outlook for Android. I am using Outlook for Android for calendar events and e-mail handling. As I can't edit contacts on Outlook for Android I am doing it on built-in contacts app. There is not much difference between functionality of Outlook
contacts and built in app on the phone. Everything seems to syncing everywhere correctly. Tested creating, editing and deleting on PC-s, Outlook.com website, phone's both built-in apps and Outlook for Android app. It's strange why the contacts are read only
on Outlook for Android as calendar works just fine.
I first created Exchange-Activesync account on the phone and turned all syncing on for it. Then I installed Office for Android and added the same outlook.com account into it. Syncing contacts is turned off because it created double contacts on phone and
messed up contacts on Outlook itself. After I turned off contacts sync from Outlook, both contact lists (phone's and outlook's) got corrected by itself.
So I guess Outlook for Android and phone's built-in contacts and calendar sending/getting data through outlook.com servers and not directly inside the phone's OS. I have also gmail account which I'm using on phone's gmail app. Google calendar and contacts
seems to be working normally as well, not using these so much.
I am quite happy everything started to work on my new Android phone. After being windowsphone enthusiast for years I finally had to throw my Lumias away and bought Android. Finally something which working correctly and as expected. Until MS will mess-up
outlook.com accounts or his Android apps.