In this scenario I assume the Postfix machine represents the corporate SMTP server. If so, it doesn't need to "listen to other computers on the network" in that sense. It already listens on the typical SMTP ports (25, 587, 465, etc depending on configuration).
What you need to look into is how to create a milter (mail-filter). Milters are inherited from Sendmail, since Postfix was designed to be compatible with Sendmail. See
Sendmail/Postfix Milters in Python[
.^]
No, you should be able to send mail to gmail without a server certificate. See
How to use Gmail as your SMTP server.[
^]. Using certificates between mail servers is good practice, but it will cost you ~USD50-150.