If you're trying to see if a slot has been booked, you can query it by ID - you will get back the status and if it is BUSY then you know the slot is no longer available. If you are unable to book an appointment and the slot still says that it is free then there are other things which could be preventing the appointment from being booked like duplicate appointment checking, patient conflicts, etc. Since our Slot search doesn't take in schedule.patient on the searching, we're not able to filter out results specific to a given patient. You might try an appointment search for the patient ID to see if they have any existing appointments at the time you tried to book the appointment.
Your third bullet point is because you used an patient id that did not exist. I'm not really able to tell much from the second x-request-id other than the appointment book did not succeed.
-Richard