Is it possible to transfer the CF9 license from the existing web server to a new one?
You cannot purchase CF9 anymore. But if you already have a license key you can just install CF 9 on the new server and use the existing license key. Unless you are not planning on removing the old server, you can only use the license in one place. In which case you would need to purchase CF11.
Your app wont just completely shut down moving to CF11 there just maybe somethings that are not supported or need tweaking to make sure its fully compatible. Depends on the scale of your application.