For the MobileMe account showing in the Accounts window of Mail preference, since it has an iCloud server listed in the window, but none in the edit list, that tells me it was created properly by the iCloud integration. I'm not sure how it works in Mavericks now. Things may have changed. The only way to remove the account from the mail list in Yosemite is to log out of iCloud completely from the iCloud pane in System Preferences. It will remove the account from mail's account list. When you log back into iCloud, don't enable Mail in the checklist and it should not reappear.
It really shouldn't be a problem just leaving it in the list disabled. Just bugs me that you get a server failure to this with Connection Doctor.
For the Cox account, it didn't work with the new user account you tested it in. Therefore, it shouldn't be a problem with your user account.
It wasn't a global permission issue.
You see the activity window processes going through the motions when you hit get mail, so that seems to be working. Connection Doctor shows you are connected to the servers OK.
Since you started having problems with it in Entourage and can't get it to work properly with Outlook or Mail, maybe it's just an issue with COX and not with your computer.
If it were a problem with something corrupted in your computer, you would need to reinstall the OS which will be a bit of work to do. It would be easier to test by setting up the account on another apple computer before going through all that hassle of reinstalling the OS which may give no results.
Upgrading to Yosemite instead of reinstalling Mavericks may solve the problem is it is something in the OS, but I wouldn't do that unless I really wanted to upgrade the OS.
Don't know what else to tell you except hopefully someone using Mavericks and familiar with Cox email will chime in with some suggestions.