If the divorce is between two Chinese nationals, then, from what I hear, in most situations, it is the mother that obtains custody of the child if the child is less than two years old. Children between the ages and three and nine are generally given to the mother, unless the mother leads a questionable lifestyle. In the case of a child over 10 years old, the wishes of the child are taken into account by the court before a final determination is made..  This is only what I have been told by my Chinese friends.  If the divorce is between a Chinese national and a foreign parent, then I have been told that in most cases custody is given to the parent who is a Chinese national.  Is the child stigmatized -- by whom and for what reason?

I've seen a few divorces and in every case where the child was a boy, he went with the father.  the one girl child i saw went with the mother.  I'll let you decide what that means.   of course the fathers had more guanxi.

That all depends. It could be joint custody or sole custody or if either parent is not fit to take care of the child the child usually goes to the more competent grand parent.