What is the ruling on marrying for the sake of the dowry, then fabricating problems to push the husband to divorce, and must the husband pay the dowry in this case?

This intention is bad and entails the sin of deceiving the husband. The dower becomes obligatory upon her by her enabling him to consummate the marriage. It is not permissible for her to seek divorce from him without harm, and he is not obligated to divorce her. He has the right to retain her until she ransoms herself by returning the dower to him, because the defiance (nushuz) originated from her, and due to Allah's saying: "And it is not lawful for you to take anything of what you have given them, unless both fear that they will not be able to keep [within] the limits of Allah. But if you fear that they will not keep [within] the limits of Allah, then there is no blame upon either of them concerning that by which she ransoms herself. These are the limits of Allah, so do not transgress them." (Al-Baqarah: 229).