Did you try going to the document on the 300 and deleting it from there (not plugged into the computer)? I can delete anything right on my 505. I never delete it from the library or use the library to delete items on the 505. I don't use the sync function either though. Have you tried Calibre for transferring documents?
You cannot buy books from Sony without the library, but there are a lot of online stores where you can purchase ebooks that will work with it. But it is nice to get the freebies from Sony if nothing else.
The MR wiki has links to many stores where you can purchase and get freebies:
MR members have also uploaded many free classics (after diligently fixing them): https://wiki.mobileread.com/wiki/E-book_uploads
By the way Welcome to Mobileread nortstudio!