Hi ALF, your 3 Apple ID's (accounts) need to have individual email address's.
@mac and @me are technically one of the same so don't separate them both will count as one, not two
for your 3 apple accounts, have @mac as your main email address, and @me as your alternate email address.
The other 2 apple accounts with unique apple ID's will need to have their own unique email address.
AND all need to be verified. Once they have been verified.
(I would delete all caches and cookies becasue that is the way I roll)
Then try to login with one account, Apple may ask for you to assign a user id, too make a unique apple user iD for each if you can (you wont be able to do this for the @mac and @me email address)