I've seen this error before and although it could be various things in my case it was always that SQL Server was not listening on shared memory. The fact that your server is not showing up in the dropdown list when you register the server leads me to believe it could be the same issue.
First of all check the error log. right at the beginning of the log, after all the databases have been started up you should see a line that reads like this:
SQL server listening on TCP, Shared Memory, Named Pipes.
When you restore in QA try and perform the restore but do NOT specify the WITH STATS option. The output of stats is performed using shared memory - in EM it uses stats by default so this is why it's always failing in EM.
If you have established that it works when you don't use the STATS option then it's almost certainly shared memory. All you can do is restart the SQL Services although if I'm not mistaken you might have to restart the server itself.
Hope that helps,