First of all you have to install the ADJ ASIO Driver.
You find this on the CD that comes with your controller or here
Maybe you already installed that, because you see ADJ ASIO as sound driver.
Now set the Sound API to ADJ ASIO, this is the very first line in the sound hardware settings.
The controller shows up as ADJ ASIO where you set the deck to channel settings.
Deck1 - channel 3-4
Deck2 - channel 5-6
Deck3 - channel 1-2
Deck4 - channel 7-8
All other connections must be set to NONE
When you finished that, you should get sound out of the VMS master channel when you start a track on your screen and open the line faders.
This is independent from the MIDI settings, you just connect here the sound of the software decks to the VMS internal soundcard channels and to the Line faders.
This is very special for this controller because the mixing is done in the controller hardware.
Next step is the MIDI connection,
When the VMS4.1 appears in the list the computer can see the controller as a MIDI device.
You write you mapped the controller,
do you mean you selected the VMS4.1 mapping from the list ?
Did you set the activated checkbox ?
You should connect the controller before you start your computer.
From time to time I have to restart the computer before the MIDI connection works correct.
Which Version of MIXXX do you use ?
Which OS do you use ?