I could enter gmail to solve the playstore-problems discussed elsewhere.
And i couldnt enter gmail at all. Not its clear:
I had set 2way-authentification through smartphone(i think it can be deactivated).
So at every login to gmail one must have its smartphone/phablet present and online, WLAN etc.
If phonenumber is entered and not gmail-adress(if having multiple accounts) it seems the first set account will be used. at least this simplification if one has forgotten his email-adress.
Very relieved now!