No, not really "Outlook 2016" - they are in the 'name every email client Outlook' stage right now but honestly, the only thing the same is that they do mail, calendar, and contacts. The phone versions are very lite clients and don't offer half the features you'll get in the desktop version.
1. OK - I guess the real term is Outlook Connector - I did it on my W 8.1 pc so I could sync to dot com on my phone - but it was a hassel and my wife was cursing all the time " Outlook - Outlook - whats the problem why do I have all these mail addresses "
Believe me I did not need this - so I just stopped it when we went to W 10.
Now we can just throw away our WP phones or use a third party that is less confusing.
2. Exactly my point - I do not want a lite Outlook on my phone at all if I could choose. I have at least 10 different Contact folders for all the organisations I and my wife are part of - OK maybe I can do it through making Groups - don´t know if dot com can handle that. Something to test. Never tested sharing our calendars either on dot com.
You are most expert in this - the other 90 % find it frustrating.
Just a button in Outlook - " sync with my dot com "
Unfortunately, can't phones can't handle the full capabilities of outlook. Tablets which support the full office suite can run it - Surface RT, Dell Venue Pro, etc - not the current phone hardware (both processor and storage space) can't support a "rich" or "heavy" email client.
As long as your 'dot com' account supports POP3 or IMAP, Outlook (or any email client) can sync with it. If it can't the problem is with the server.