Re: Office 2010
not sure what version of exchange like functionality Office 365 has but I have outlook 2010 on windows 7 connected to office 365 no issues., have been since at least 2013 I think, prior to that the company I am at was using hosted Exchange at Rackspace and it worked fine there too.
official support looks to conclude in Oct 2020 (updates etc). Haven't heard/seen anything saying when or if outlook 2010 will stop working with office 365 exchange.
Most of my email is done via Outlook web access from Linux but I do keep outlook running 24/7 in my windows VM sometimes it is useful.