Is it permissible for the wife to refuse to live with her husband in the place he has chosen (the countryside), or is the husband obligated to provide housing in the place she desires?

It is obligatory for the wife to reside wherever her husband resides, and she is not permitted to refuse to move with him, unless she stipulated before marriage that she would remain in her own home, in which case she is allowed to do so. If you did not stipulate that he move, then it is obligatory for you to obey him by residing with him wherever he wishes, as long as he provides you with suitable and safe housing.