Michael Reed, Jon Heberlin, Trevor Seward let me try and answer both of your questions at once! For context
While these older Office clients won't stop working in 2020, but client connectivity with Office 365 (such as Outlook 2010 connectivity with Office 365) will no longer work and will no longer get security updates. For context, it is good to understand what is going on with Office 2007 at the moment - once Office 2007 is out of support, we will no longer support it connecting to Office 365 and will retire older protocols, such as RPC/HTTP. Unlike Office 2007, we are still 3+ years out, so not quite ready to talk about which protocols will change and how - but as we get closer, we will publish support articles with more detail, provide targeted admin notification through message center and closely watch our service telemetry to ensure we are minimizing impact to customers.
What is changing is 2020 is what we consider an older client. Today we make no explicit statement of when a client will stop working other than to say the experience will degrade over time and we will not support anything that is no longer supported. Going forward, we want to get rid of the ambiguity of "degrade over time" andexplicitly define the clients that will work as those in mainstream support (ie, released in the last 5 years) and the Office subscription clients that are always up to date (Office 365 ProPlus or Office 365 Business). 2020 also marks the end of support for Windows 7.
What this means for admins supporting environments with Win7/Office 2010 is you should be taking these end of support dates into account when thinking about your longterm desktop strategy. We are providing 3 years notice of this change, so you don't have to move those clients immediately, but as you plan your desktop strategy over the next 3 years, you should take this change into account. This may mean planning to deploy Office 365 ProPlus (or Office 365 Business), using Outlook on the web or deploying the next perpetual Office client release which I referenced above.
If Office 365 ProPlus is your choice, FastTrack has engineering resources to give you guidance, best practices and help you get up and running, at no additional cost.