Usually, not much can go wrong with a simple and polite 'Thank you' - Email. You'll usually not burn any bridges, unless your mentor is some kind of jerk, and most people will see it as the polite way to handle things.
If you want to stay in contact, mention it. Just accept it if your mentor decides otherwise.
Why do you need to leave to try and get an offer elsewhere?
The reasons for them not providing you with an offer are not particularly relevant, although none make it a good place to work from a career perspective.
They don't want you permanently, but need you for now and are stringing you along
Your manager has only limited power in the organization
But if the interviewer asks what would be my plan upon the completion of internship, how should I properly answer this question? Should I not mention about graduate studies at all?
I think you should be honest and tell them about your studies (as that shows interest in learning), but that doesn't mean you are closing the doors for a full-time offer by ...