Proper naming of gateway host

["Gary O'Neill" <gary@hedgehog.dhs.org>]

I've been curious about this for some time now:

My gateway machine acts as a mail and dns server and performs NAT.  It
is known in dns as hedgehog.dhs.org to the outside world and as
isthmus.xanth internally.

As the box can only have one hostname, which should it be?
It currently considers itself to be hedgehog.dhs.org.
Also, which hostname should the qmail 'me' file be set to?

Thanks,

Gary

PS:  On a tangent, what is the auth port (113) used for?