Domain name issues.

[Rasjid]

Work has a website hosted with TPG.
PING austethical.com.au (203.26.24.101) from 192.168.0.2 : 56(84) bytes
of data.
64 bytes from virtual8.tpgi.com.au (203.26.24.101): icmp_seq=0 ttl=111
time=805.121 msec

We are in the process of getting our new E-Smith server to do our
internal mail.  Currently all our mail goes through TPG as well. 
However, our internet connection is through Telstra.

What we want is any mail addressed to austethical.com.au to be delivered
locally.

So my question is:
Can we set the name of our server to be server.austethical.com.au? (It
is behind a firewall and does not have a public address.)  Would this
cause a problem, even though www.austethical.com.au would be on a
completely different network?

If this is okay, then presumably any mail sent to the server with
address austethical.com.au would be delivered locally, and our problem
is solved.  If not, then I will have to read up about Qmail.

Cheers,

Rasjid.