Apple ID says my googlemail account doesn't exist

I have two ipad 2's and two iphone 4's linked to the same Apple ID account and all 4 devices have been unable to link to the Anything Apple (iTunes, iCloud, App store etc. for over a year because Apple says the email - a googlemail account - doesn't exist. I get email from Apple several times a day to this account and I get a message on all 4 devices asking for the password for the account but when I enter the password or try to reset it says the account doesn't exist. I have spent about 20 hours total online with Apple and 4 hours twice at 2 different Apple Stores trying to get this problem corrected. One of the stores changed the email to gmail....same account...gmail and googlemail addresses are interchangeable, or at least mine is, but it still says the account does not exist. I was willing to simply lose everything on all 4 devices and simply re do them with a new apple id but they can't do that either because they say they do not know that I own them despite having receipts in my name for all 4 devices. I can't download aps at all with any of the devices and they are all virtually useless. Apple seems to be at a complete loss and has suggested that the best thing to do is probably replace all 4 devices....not a bad idea considering their ages I admit but also no small investment. They work fine now but at some point they are going to need a software upgrade that is not possible with no access to the Apple Id. The inability to download apps is nearly debilitating for the devices as the entire Apple world is an app. Any suggestions?