It depends on the registrar, probably. If you install DHCP IN YOUR network You must be change the dhcp IP range in dhcp proprties that is asing by DHCP, OR use the netdom cammand to change the computer name of the domain. i can change the domain name.
Yes, if an IP address is changed, the associated A (or other necessary) records should be changed. Otherwise there is a disconnect between the name and the address of the host.