Is the husband's behavior — wherein he refrains from providing a separate residence for his wife or allowing her to visit her sick friend, refuses to care for her son from another man, and prevents her from studying and receiving an allowance, while permitting his other wives to do so — considered just, and what is incumbent upon the wife to do?

It is the wife's right that her husband provides her with a suitable, independent dwelling. She is not obligated to live with her co-wife. It has been reported from Ibn Qudamah in "Al-Mughni" that it is forbidden to house two wives in one dwelling without their consent, due to the harm of enmity and jealousy. It is preferable for the husband to allow his wife to go out to seek knowledge if she needs it and he is unable to teach her. As for the maintenance (nafaqa), it is due day by day, and giving it weekly in advance is not obligatory. The wife is not permitted to leave her husband's house without his permission, except in cases of necessity.