Are the servers in the same AD site? Generally you set the SCP to the same URL so that you can use the same certificate, but there is no requirement to do that.
As long as the cert is trusted the clients will use it w/o any errors and it will work just fine. They wont stop and switch over. The Outlook client may pick up any of those SCPs in AD
And yes, you need to recreate any custom receive connectors on the new server if required. You can export the settings for the original using Powershell them create the new ones with the same settings except for any IPs bound to the server itself.