Ok, tested a newer game (since the other ones seemed to be developed ages ago) called The Last Remnant and it works perfectly!
But when I tried to play Fantasy Tennis (also a game from Alaplaya like S4 League) and it didn't work - I got the black screen.
A little bit more information:
After I got the black screen I pressed ctrl-alt-delete and then a message popped up:
"Failure to display security and shut down options"
"The logon process was unable to display security and logon options when CTRL+ALT+DELETE was pressed. If the operating system does not respond, press ESC or restart the computer by using the power switch.
And whatever I do (press ESC or OK button), the alert goes away, but I still see the black screen...