I would suggest you are over-complicating. You can easily keep all of the information in one table. (And certainly not another database.)
You can have a short-order layout that collects all the basic info required up to the point of their starting appointment. If they then get tagged as 'Started' you can add the more detailed information on another layout, and even related records of subsequent appointments, for example. If ever you find yourself saying something like 'I enter a lot of duplicated data', or 'I have a table that is almost exactly the same as another table' then alarm bells should sound that the design should probably be reviewed.
Okay, so the problem I'm running into with one table occurrence is the following: I keep track of two main statistics that are related, New patient exams and New patient starts. All new patients have an exam, so it's easy to do statistics on the information such as whether or not they showed up for their exam, how old they were at the time of the exam, etc.. I run into issues with the New patient starts, however, if I keep it all in one table occurrence. Not every patient that has a New patient exam starts, so I have a lot of records with empty data. That is problematic for getting back averages on important stats such as Fee charged, Discount given, etc. Any ideas on how to improve this? Is there a calculation that I can do to rid myself of all the blank fields without them being included in the calculation, or should this information be kept in separate table occurrences?
A table occurrence and a table are not the same thing. Tables are created on Manage | Database | Tables. Table occurrences are the boxes created in Manage | Database | Relationships.