As far as a catch all post-upgrade checklist I do not know of one currently. That being said you can review all release notes here :Release Notes | Tableau Software. This will show you all the features that are being changed in the coming release or that have already been implemented.
As for a specific issues I would need a little more detail on the rendering issues or connection issues that occur post-upgrade to advise how to prevent these things from happening or if they will need to tested in the future.
Additionally, if you are running a server instance, here is the pre-upgrade checklist for Tableau Server :http://kb.tableau.com/articles/knowledgebase/Pre-Upgrade-Checklist?lang=en-us .
Let me know if this helps or if you have additional questions.