Q1. In my university studies of object-oriented modelling and design they recommend thinking about what an object can do for its method, and what its responsibilities are for it
Q1 Is it possible responsibilities of your objects should be interpreted as authorization or contract requirements, as in what actions they should take? So, to take a medical example from your Q2, an object with a Scheduler role (think C#/Java interface, or Obj-C protocol) has attributes around CanEditAppointments or EditsAppointments.
Q2 From a use case perspective, a patient may be able to create an appointment, and so you might implement a Patient in your object model with a method to CreateAppointment(). But, in terms of encapsulation, you would likely instantiate an Appointment object in CreateAppointment(), and then call methods or set properties on the Appointment object to set its time, date, patient, physician, etc.
And because the Appointment collection is likely to be permanent storage like a database, it would likely be the Appointment object's responsibility to add itself to the collection (Appointment.Schedule() goes through your data access layer to save itself to the database).
That also ties back to your Q1, in that the Appointment object's responsibility is to save itself, so it might implement an ISaveAppointment interface that requires fields and methods to carry it out. It also is the Appointment's responsibility to have a date, and time, and patient, etc., before being saved, and so the ISaveAppointment interface should require they exist, and Appointment.Schedule() should validate the values are correct or have been previously validated.