Since you never mentioned the platform, all I can offer is advice for Windows. 1-(the easy one), for the Yamaha kit, disconnect the midi cable going from the interface midi OUT to the DTX midi IN(now you can't cause a feedback loop). The only reason to have that cable connected would be to use the sounds in the DTX brain).
2-to prevent the Korg from "driving" the drum sounds, go into Setup>Midi>Midi Studio and do some small changes. This setup window should show 2 "instruments"(the Korg and the Yamaha kit). for the Yamaha kit, set the midi INPUT PATH for channel 10 only(channel 10 being accepted as the usual channel from drum/percussion use). The output channel can be set for 10 only, or don't select any(if you don't need to send midi back to the DTX).
3-for the Korg(in the same setup page), select all channels EXCEPT 10.
4-Go to VIEW>Mix Windows Shows> and show Instrument. THis opens a row across the top of the MIX window. On your instrument tracks, you can select which controller(Korg or Yamaha) for each track. Doing this should allow the DTX to trigger ONLY drums, and the Korg to trigger anything else
(by default, Midi and Instrument tracks will receive midi from ALL, which means any incoming midi).