Postfix, Exim, Sendmail and the rest are Mail Transfer Agents and can act as a Mail Submission Agent, Mail Exchange, Mail Transfer Agent, and Mail Delivery Agent. Given the proper cofiguration they will both send and receive email. (If you want to run your own Mail Exchange (MX) server you need a fixed address, and should get your IP address provider to configure your PTR address with your mail servers domain name.)
However, to retrieve email from a service like mailinator you need a program like
fetchmail which will read mail from an inbox and pass it to your local MSA (Mail Submission Agent). The trick with a service like mailinator could be to configure pick up addresses dynamically.