DNS possible for private networks?

[David C. Hoos]

Here is what I want to do, but am not sure whether it can be done:

I have a RedHat 9 box with three NICs.

One NIC is on our corporate LAN behind a firewall.  Through this
interface we can browse the Internet and allow telnet sessions from
various workstations on that network.  DNS service on that network is
provided by a DNS server which is part of our corporate network, and
in turn from our ISP's DNS servers.

The other two NICs are for private networks, viz.

   10.0.0.0/8
   172.16.31/16

The computers on these networks only need to communicate with the
others on his own network, and with the three-NIC Linux box.  The
three-NIC box provides no routing services.  They will not be
accessible from any other network.

All of the computers on these networks are configured for DHCP, and
the DHCP server on the three-NIC box serves fixed addresses based on
the MAC address and name assigned to the particular computer.

What I want to be able to do is to provide lookups by name of those
fixed  addresses to users on the three-NIC box, as well as to all of
the computers on the private network of which they are a member.

I have the following questions about this, viz.:

  1.  Can I do this?
  2.  How do I do it?
  3.  Can I use my own made-up (i.e. unregistered domain name -- e.g.
domain1, and domain2) or must I use a registered domain name?
  4.  If I must use a registered domain name, can I use our corporate
domain name?
  5.  If this is doable, is there any way to have a common database of
hostnames vs. IP addresses for use by both DHCP and DNS (BIND), or
must I duplicate the information for both?
 
Thanks for any illumination you can provide on this.

David Hoos