Does this make sense? Or is there anything we should be doing to prepare Site Kit before the DNS change?
When you mention a DNS change it sounds like you’re moving a domain name (temp.com) to another provider. If your new domain (mysite.com) is already pointing to the correct host, and in turn – new site, you don’t need to make any DNS changes. Is only a matter of installing Site Kit on this new domain. As it sounds like this is the case, with you mentioning so above, have you tried to install Site Kit on this new site?
One other question. Is there anything we can do to make setup easier on the new website? Any way to clone or record all settings from the old one?
When setting up Site Kit on a new domain, Search Console will need to reflect data based on the new domain. Site Kit takes care of this for you by connecting your site with Search Console, creating a property for the new domain or using an existing property if you already have one for this domain.
In the case of Analytics, you can use the same previously connected property on the old domain if you wish, in order to keep new hits recording to the same property.
As for AdSense if applicable to you, unfortunately users need each domain name already approved before ads will appear. Data such as ad performance is not transferable from one AdSense site to another.
Let me know how you get on with the site move, or ask if you have any further questions.