To what extent is a husband obligated to care for his wife's family? And is a wife's departure from her husband's home after he insulted her mother (despite his apology) considered disobedience to Allah's command to obey her husband?

A husband should maintain ties with his wife's family and treat them well, as this brings joy to her heart, and it is part of the good companionship enjoined by Allah. The wife should know that obedience to her husband takes precedence over obedience to her parents due to the greatness of his right over her. It is not permissible for the husband to prevent his wife's family from visiting their daughter unless he fears corruption from them.

As for a wife leaving her home without her husband's permission and traveling without a mahram, this is forbidden by Islamic law.