It's important to understand that the service applications have their own set of databases, which is seperate from the content databases. That being said, the answers to your questions are:
The service apps databases will remain on the old SQL Server and will continue to function from there. If you wish to move your service apps databases to the new server as well, you'll need to backup and restore the databases from the old server to
the new server and then recreate the service apps using the existing databases that were restored to the new server.
You do not have to stop the SharePoint farm, especially in cases where you have multiple content databases. When you scheduled outage window arrives, simply detach the target content database from SharePoint either via the UI in Central Admin or using
Powershell. Backup the database on the old SQL Server and restore it to the new SQL Server. Now attach the content database using the UI in Central Admin or Powershell while pointing to the new SQL Server and all content should come up without
any issues.
