Jushiro, I think you would be best helped by picking up a book on PHP/Mysql...
You don't actually store the PDF file IN the database, there is merely a reference to the file name in there (and sometimes even the complete file path, if your aren't worried about bit size in the DB).
1. You are missing the primary key column for the database.. if you don't know what this is, i highly recommend reading up on the basics of MySQL
2. Use a standard upload form as BorderLine suggested to upload the PDF (or any other file)
3. Retrieve the file (as thorpe said) by using a standard anchor <a> "link" tag. the $_FILES array is used when uploading files from a webform to the server and has nothing to do with the database processes
4. use the primary key as the connection between your select menu and the appropriate file
eg:
<select name="teachers-docs">
<option value="primary_key_goes_here">Document for math homework</option>
<option value="primary_key_goes_here">Document for science homework</option>
<option value="primary_key_goes_here">Document for history homework</option>
</select>
when the student selects an option, a query can be submitted to the database calling for the file_name column for primary_key # x which will give you the appropriate file name.
I recommend using $_GET to submit your form, as it will make testing and implementation slightly easier.
Also, provide a link to the file and allow the student to download the PDF (to make things easier)
good luck