Does the Xone 23C work with Traktor

No it wont.
Native Instruments has stopped the “Traktor certified” program as of this year and hence no more hardware that is non NI will be supported.
The X32c is actually made for Mixvibes Cross and comes with an LE version of it.
You can of course use the X23c with Traktor, however not with the built in soundcard, you would basically use it the same as the X23 (that has no soundcard) and plug it into an NI soundcard like the Audio 6.

So no to both of your questions.