My thoughts without reading the other responses:

Is it the employee or the managers responsibility to make a time for monthly supervision? I'm not quite sure what is involved but my understanding of the word "supervision" would indicate that it's the manager's responibility.

Is it the role of the manager to remind employees to submit monthly reports, or should staff be responsible to ensure they are done? If it's the role of the employee to provide reports, then they should do so as a matter of course.

Should the manager send out a reminder for when time sheets are due? No as above.

Is it the responsibility of the manager to approach staff to make a time for a performance appraisal, or should staff be responsible to approach the manager to make a time? Either. Managers should ensure it is done on a timely basis as per the contract (for me this is annually). But if employees have issues or situations they need addressed, then there is no reason why they can't approach a manager for an interim meeting.

ETA: Apparently I agree with Efjay too
ETA2: As both employee and manager, it really annoys when I have to remind people to do things they know how to do and it also annoys me when my boss reminds me to do things before they are due without giving me a chance to do it first.