- Google Docs doesn't work with PDF files. It has its own file format. Do you mean Google Drive, perhaps?
- Yes, you can set up a submit button that will send the file to your Gmail account. Add a button and set its MouseUp action to "Submit Form" and as the target URL enter "mailto:firstname.lastname@example.org" (using your own email address, of course).
The user will have to have an email client set up in the application, though, and will need to manually send the email, though. It will not be sent automatically the moment they press the Submit button.