After you change the dns to the new server you must do a rebuild to get let’s encrypt to issue a certificate. If you rebuild without dns in place you risk hitting rate limit issues.
If you have access to the old server and it has certs then you can copy those over. See Move your Discourse Instance to a Different Server
It is, but it’s a little tricky. Change the domain name or rename your Discourse.
If you do a restore of a backup it’ll do that remapping of the name. I’d recommend using www rather than the bare domain. You can use forcewww.com to redirect to www (there are many other ways). In chrome, users can’t tell the difference between the bare domain and the www subdomain. If you want to Configure direct-delivery incoming email for self-hosted sites, using the www will make things easier if you already have incoming mail set up.