OK, so there are a few considerations here.
If you are going to migrate everything at once you can certainly install replica Connection Servers in the new site to have the data replicated over. Once the new Connection Servers are up you can simply uninstall the old Connection Servers from the original site.
You can't run the old Connection Servers and the new ones in production as you will run into issues. You won't corrupt the database as long as the link is stable during the time you are doing the install/uninstall.
There is no such thing as a 'primary' Connection Servers. They are a cluster and all operate on the same level so there is no need to do anything special once you uninstall the old servers.
For backup, just backup the database of each component. View does an automatica backup each night at midnight by default so you can just grab the latest files. This includes an AD LDS (formerly ADAM) backup as well as a View Composer database backup. The vCenter database and Composer databases can also be backed up using the built-in SQL backup.
As for accepting connections, technically View could continue to accept connections but you may run into some connectivity quirks as View isn't designed to operate in this manner. If it is a very short window then you should be OK.
Let me know if you have more questions or if anything is unclear.
-Mike