Replace domain controller/data sharing server

You won't be able to have both servers running with the same name.

You'll have to bring up the new controller with a new name.  You'll want to move all the DC roles over to the new server once you've promoted it.

To copy the files, I would use robocopy.  You'll still have to recreate the user shares and the rights on the share folders.  Either way, once you have all the files copied, You'll want to go back and select the shares, go to properties, advanced, and replicate/replace permissions on the files and sub folders of the shares to ensure they have the access rights you desire.

Once everything is working - AND - the dc roles are on the new box, demote the old box.

Also, you'll want to see if that old box is doing other things such as DNS, WINS, DHCP for your environment and setup the new one correctly for that.

On the server name - share problem, perhaps setup an Alias record in DNS once the old server is gone, pointing to the new server IP.

This is a multiple step process. First things first, I would reccomend against trying to do this with the same name. We can handle this later with an ALIAS.

1. Active directory - First youll want to get your new server all built with 2003 etc and ready to be DCPROMO'd. Youll want to dcpromo the new server and make it a global catalog server. Now you should have 2 domain controllers on your network.

2. Transfer FSMO roles (if any) - in your question you didnt mention if you had other DC's so im only assuming one. You will want to follow these directions to find and transfer the 5 FSMO roles to the new server.
http://www.petri.co.il/determining_fsmo_role_holders.htm
http://www.petri.co.il/transferring_fsmo_roles.htm

3. Demote old domain controller - Once you have confirmed that all fsmo roles have been transferred and tested everything you can then demote the old DC to a member server. After a reboot it will just be a member server again.
http://technet.microsoft.com/en-us/library/cc740017(WS.10).aspx

4. Copy the DATA to a new server - for this I would reccomend using Robocopy with the /mir /sec /copyall command. This will copy all your files and folders with their existing permissions.
http://technet.microsoft.com/en-us/magazine/2006.11.utilityspotlight.aspx

5. Recreate shares - Robocopy cant create the shares and sharenames for you so you will have to remake these yourself. I would make the shares the same to reflect the old servers ones. This way when you create the alias the paths will be the same.

6. Remove old server from domain and make alias - Once you have confirmed AD and the data shares all look good you can remove the server from the domain and delete the computer account out of AD. Oncet this is done you can create an alias record in DNS for the new server with the old server name. This way when the user types //oldserver/share it automatically directs them to //newserver/share.