What is the ruling on a wife remaining without a niqab and refusing to bear children after repeated requests from her husband, and is it permissible to take her back after divorce due to these problems, given that there is a child between them?

Covering the face is obligatory for a woman, and it is more emphasized for your wife if you command her to do so. Procreation is a right for both spouses; neither has the right to prevent the other from it without a valid excuse. Therefore, if your wife refuses to procreate without a valid excuse, or if she refuses to cover her face—in the case that it is obligatory for her—then she is defiant (nashiz). The treatment for defiance follows specific Sharia steps, and divorce is not the first among them. Divorce should only be resorted to if its benefit is clearly outweighed.