I recently restored my 2013 MacBook Pro El Capitan system back to Mavericks from my Time Machine external drive. The restore was successful except a few that wont start: Calendar, Terminal, Messages, App Store. I tried loggin in as guest, same problem.
If these were 3rd party apps I'd say that you need to reinstall them but they aren't so it's a bit... odder...
It may be that the Time Machine restore put back files that were created with a later version of the app that the older version understands. I would start with deleting the preferences files for those apps in ~/Library/Preferences. Look for files that start with "com.apple" and end with the app name. You could also check in /Library/Preferences. Delete the preference file(s) and try the app again.
Apps like Calendar have a file that they automatically open. you may have to just delete the primary calendar 'document' which (I believe) have their own folder in which they reside: ~/Library.
Finally clear the caches folders Delete the contents of /Library/Caches and ~/Library/Caches and reboot.
At least that is my best guess.