I assume I will need to point MX records at my server (LAMP), -- what processes the incoming e-mail message?
Are there any existing PHP libraries to do this?
I would recommend you to do processing in Perl (python is also ok, but Perl has very similar syntax to PHP), which is much more suitable for the task. You can also find a lot of libraries through CPAN there.
http://search.cpan.org/~rjbs/Email-Simple-2.100/lib/Email/Simple.pm