There is a way. My sysadmins do it every time I've moved over the years. I believe they copy the database to the new server and on the old server set the new server as the database for it so even if their dns hasn't updated they are still connected with an updated database.
Again. ..i think that's what they do. You can always contact admingeekz to help you.
Copy all your files/databases to the new server, and on the old server set the database details to that of the new server (you may need to allow remote mysql access). Now the old server is using the database of the new server, so the content will be the same.
You just have to make sure not to install/update any add-ons until the DNS propagation is complete...
Shouldn't. It's going to make a difference in how you transfer the files and how you set up your new site, perhaps, but DNS is DNS. As was mentioned above, change the TTL, set up your site on the new server, once you have made sure everything is working on the new server, change the DNS settings to point to the new server. There should be no downtime involved at all.
If you use Cloudflare to manage DNS, they are really quick.
Plus using the methods suggested by others pointing the database to the new server, and also use rsync to have a complete copy of every files .
From my understanding, AdminGeekz creates a copy, moves it over to the new server, all while the site is still up and active. Then runs something to update the new server database to that of the old, then points to the new database from the old server...
I could also be making all of this up When it comes to server stuff, I hire out. If downtime means that much to anyone, they should hire a specialist to move servers without any downtime, which I know AdminGeekz can do.