Determine whether mixed mode auth is enabled without logging in?
You can check it in the registry:

HKLM\Software\Microsoft\Microsoft SQL Server\ [instancename] \MSSqlServer


The current mode is recorded in the LoginMode key.

Values (DWORD) can be:


1 = Windows Authentication mode
2 = SQL Server and Windows Authentication mode


Note: 


0 is equivalent to 2
You still need access to the Registry (and the server)