1) If your sshd config files are stored under /etc/ssh/, you'll need to copy all of those. You'll find the sshd config there, as well as the host-keypair.
Make sure you correctly copy the permissions too! The sshd will simply ignore keys that are not properly protected. (Good as well, as an unprotected private key is a very bad idea.)
2) If you have already added public keys of remote hosts that are trusted and don't require a password to log in anymore, you'll also have to copy that info in order for the auto-login to work again now. This info is stored normally in /home/-account-/.ssh/ (under authorized_keys). Again, don't forget the permissions here.
Regarding the IP address change, assuming the remote clients actually see a change in IP (you're not behind some reverse proxy or anything), then no, they will require the user the accept the certificate again. (since in the known_hosts file of the remote client, the server's IP & server's certificate were stored together) Worse, if you assign that IP address to another ssh server, they might even be warned of a man-in-the-middle attack.
So just using the same (external) IP address on the new server makes sense.