Its not possible to use ASIO4ALL and your onboard Realtek soundcard at the same time.
To be quite honest, this will create havoc and driver conflicts on your set up and produce some undesirable results which I have experienced in the past.
You should however be able to select your devices in preferences and have them all use the ASIO4ALL driver.
im not really understanding why that wont work.
However, I'm guessing what you might be seeing is the effects of needing to update to a better Soundcard that is capable of both audio and midi monitoring. They are not expensive and a Focusrite Scarlet will get you in the ball park for around 100 bucks.
The only method I see possible is using 2 computers (one for soft synths midi usb/Realtek only and use the line output from that PC and send it in to your mixing console/The Behringer.
Those mixing boards with USB Stereo line recordings capability's produce less than stellar results when it comes to monitoring and recording in multi track format. They are more often than not, used to record or monitor a stereo mix and not track by track.
I know because I have a pretty decent one (The Mackie Pro FX 16 USB). I only use it for a summing mixer or for drums (sending the Drum Mic's in to the mixer, then stereo in to my Focusrite.