Will the wife be sinful if she insists on returning to her home country and leaving her husband to complete his studies abroad, despite his need for her? What should the husband do in this regard? And what is Islamically preferable for the wife and daughter's residence: in an independent home or in her parents' home? What are the husband's financial obligations in both cases?

If you both agree to your wife remaining in her country while you study and travel, then there is no harm. However, she does not have the right to compel you to do so. She must obey you as long as it does not harm her. You are obligated to spend on her and your daughter what suffices them and meets their needs. The estimation of the expenditure varies according to your means and custom. Your daughter's expenses for food, clothing, education, medical care, and housing are obligatory upon you to the extent of her needs. It is better for her to reside in the safest place.