DDJ-WEGO + Rekordbox don't work

That second screenshot shows the audio device; scroll to the bottom of that window and see if the WeGO appears beside Control Device Information (on the Audio page of the preferences).

It does http://prntscr.com/e246hj. 

For the sake of experiment I've connected DDJ-SB2 to the same machine. The device worked just right with no settings manipulation.

With the WeGO connected, can you open the Audio MIDI Setup application, go to the MIDI Studio window, find the DDJ-WeGO, select it, then click Remove Device. Unplug it from your computer, then re-connect it and click Rescan MIDI.

Close that application and launch rekordbox, see if it works.

Well, I did what you'd told me. The MIDI studio looked like this http://prntscr.com/e3gi6q. I started playing a track with a controller + RB, checked all buttons, so everything was just fine. After 1 minute the sound suddenly vanished and the controller lost control, none of buttons worked, no sound from speakers. Then I went to Audio MIDI setup again to repeat the procedure, removed the device but rescaning couldn't find it again. So I restarted my machine, readded the controller to the MIDI studio, opened Rekordbox and it was fine again, but after 1 minute everything happened the same 😐

Have you also tried using different USB cables / ports to eliminate those as potential problems?

No I haven't, but the device works perfectly with traktor being connected with the same cable.

Can you try disabling the "Master Output" (click the purple speaker beside "Master" up at the top of the rekordbox window)?

I just gave it a try on the same conditions (WeGO1 firmware 1.09 on rekordbox 4.2.5 under OSX 10.12.2), and had no issues.

What's odd is if it's working properly with Traktor, reason would stand it would also work with rekordbox rather than kicking out after 1 minute.