What happens to the tenancy rules after the agreement ends and isn’t renewed? For instance, if the agreement expires in June, does it mean I have to move out by June 30th or by July 30th? Could you explain what the “calendar month” notice period means in this context?

Under Indian law, if a tenancy agreement isn’t renewed, it automatically converts to a month-to-month tenancy. You must vacate by the end of the calendar month following the expiry date (June 30th in your example). A “calendar month” notice period means you must give one full calendar month’s notice before vacating, starting from the first day of the month.