Change of L visa to S1 type

I have understood that I can only get a new wisa(Z) in my country of residense (Denmark) I am pressent in China and waiting for an invitation letter etc. because I have got a work in China
If I go to Denmark to apply for the Z type should I also bring my daugther or can my daugter stay in China together with her Chinese mother?
If I only bring me daugter's passport the passport will show she is still in China because of no exit stamp. Is that a problem?