This is not a problem with Outlook or your accounts. You just need to set up Outlook to keep you logged/signed in. Sometimes the security settings of the server prevent you to keep you signed in for a longer period of time and you will need to manually enter your account. On the other hand, this could also be the cause of the user permissions that prevent you from logging in. My advice is to completely wipe all the account from the Outlook and reset its default settings from the configuration menu. Once you do this, add just one account and watch the behavior.
Also, please identify MS Update KB2553248 in the uninstallation list and remove it. It has been reported as a fix for this type of problem.