Can someone schedule an appointment with me without being in my system?

Anybody who is not in your Console system yet can go to your calendar page and schedule any type of appointment that you have available for prospects or all contacts (setup as such in the appointment type permissions: Calendar tab > Appointment Types > appointment in edit mode > tab "Permissions"). People in this scenario will be required to register in order to schedule and will be entered in your system as prospects when they do so.