By short names, do you mean the NetBIOS names? If you want to prevent that from doing broadcast resolution you would need to set the node type in DHCP and/or setup a WINS server.
The MS DNS server will happily handle multiple domains (just setup a zone for the ones you want it to handle) and forwarding of other (external) domains to external DNS. But (to get the zone type correct) it would be best to know more about why you have multiple domains and what if anything they are seperated by.
Are they all parent domains, or are some child domains?
Are they all in the same forest?
Is this a LAN, WAN, MAN, or VPN based network?
Am I correct in assuming this is a Windows network...?