Some of the standard Windows 10 Apps (Calendar, Mail, Calculator) refuse to open when I sign on my laptop with my Microsoft Account. If I sign on with another Local Account, they work fine.
That would seem to indicate a corruption issue with your profile.
If you've rebooted several times and confirmed that it is not simply a glitch, then.......
You may have to create another user account as a member of the local administrator's group (not a standard user) and recover your profile items into the new account (copy them, do not move them, you can clean up later). http://windows.microsoft.com/en-us/windows/fix-corrupted-user-profile#1TC=windows-7
The process remains the same as those instructions for Windows 7. The caveat being it will require a third admin user to accomplish the task as you cannot be logged on as either the user you are copying from or to.
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