okay i came a bit closer.
i did following now:

go to database "mysql" and table "user"
found "root"
changed host from "localhost" to "%"
in shell did: "mysql -u root -p flush-privileges"
entered password
worked.
now i can connect to my database ON the pc with my internal IP, which is my external too (im at the university right now).
though i can't connect from outside now (maybe cause of the router, who knows) but im a step closer now.

EDIT:
okay, everything works perfect now. there were some firewall issues so i couldnt connect from outside, but from inside.

thanks again for this great help. the main problem was the user table in the mysql database