I made everything you requested, I think that covers what Bill wanted. There is no DNS server on the machine itself, so I skipped those parts.
cat /etc/hosts
127.0.0.1 localhost.localdomain localhost
x.x.x.x mail.mydomain.com mail
# The following lines are desirable for IPv6 capable hosts
::1 localhost ip6-localhost ip6-loopback
fe00::0 ip6-localnet
ff00::0 ip6-mcastprefix
ff02::1 ip6-allnodes
ff02::2 ip6-allrouters
cat /etc/resolv.conf
# Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8)
# DO NOT EDIT THIS FILE BY HAND -- YOUR CHANGES WILL BE OVERWRITTEN
search mydomain.com
nameserver 193.6.238.2
nameserver 193.6.238.6
hostname
mail
hostname -f
mail.mydomain.com
ifconfig eth0
eth0 Link encap:Ethernet HWaddr 00:25:90:31:21:f6
inet addr:x.x.x.x Bcast:x.x.x.xz Mask:255.255.255.240
inet6 addr: fe80::225:90ff:fe31:21f6/64 Scope:Link
UP BROADCAST RUNNING MULTICAST MTU:1500 Metric:1
RX packets:63114 errors:0 dropped:0 overruns:0 frame:0
TX packets:44889 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:100
RX bytes:38603923 (38.6 MB) TX bytes:12473239 (12.4 MB)
Memory:fafe0000-fb000000
ping
Google
PING
Google (209.85.148.105) 56(84) bytes of data.
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=1 ttl=51 time=45.6 ms
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=2 ttl=51 time=18.6 ms
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=3 ttl=51 time=18.9 ms
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=4 ttl=51 time=19.0 ms
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=5 ttl=51 time=18.6 ms
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=6 ttl=51 time=19.0 ms
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=7 ttl=51 time=18.8 ms
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=8 ttl=51 time=18.8 ms
64 bytes from fra07s07-in-f105.1e100.net (209.85.148.105): icmp_seq=9 ttl=51 time=18.8 ms
nslookup mail.mydomain.com
Server: 193.6.238.2
Address: 193.6.238.2#53
Non-authoritative answer:
Name: mail.mydomain.com
Address: x.x.x.x
dig mydomain.com mx
; <<>> DiG 9.7.0-P1 <<>> mydomain.com mx
;; global options: +cmd
;; Got answer:
;; ->>HEADER<<- opcode: QUERY, status: NOERROR, id: 22354
;; flags: qr rd ra; QUERY: 1, ANSWER: 1, AUTHORITY: 2, ADDITIONAL: 3
;; QUESTION SECTION:
;mydomain.com. IN MX
;; ANSWER SECTION:
mydomain.com. 150 IN MX 10 mail.mydomain.com.
;; AUTHORITY SECTION:
mydomain.com. 45 IN NS ns.deninet.hu.
mydomain.com. 45 IN NS ns.serverpages.eu.
;; ADDITIONAL SECTION:
mail.mydomain.com. 122 IN A x.x.x.x
ns.deninet.hu. 45 IN A 195.70.35.5
ns.serverpages.eu. 45 IN A 79.172.211.90
;; Query time: 43 msec
;; SERVER: 193.6.238.2#53(193.6.238.2)
;; WHEN: Sat Nov 5 18:35:16 2011
;; MSG SIZE rcvd: 152