Per default one limitation which a test-server had is you could not lease a licence. But you could lease on from your production environment and use therewith the test server. And of course you could contact your partner, maybe there are possibilities.
I would say not really - there must be no association. By us it's easy the test-server is named like the prod-server with the suffix of -t. You could try it with a ping-statement but better would be to ask the qv-admin.
You could check if the Test server has Remote Management configured. That's a publisher set-up that allows for tasks to be pushed from test to production (and vice-versa) as-is via the QMC. A requirement for this mechanism to work is that each system should have the other's name configured.
See QMC->System->Setup->Remote Management Services and check the right pane for any configured remote systems. With that information, you should be able to identify the production server.