Is it permissible to propose to a girl who has been engaged for eight months, and is it permissible to ask for her opinion after her family previously refused to break off her engagement?

It is permissible for you to ask a suitor for his fiancée, and request him to leave her for you. If he agrees and willingly leaves her, then it is permissible for you to propose to her.

Similarly, if she herself breaks off the engagement. However, as long as she has not broken it off, or the suitor has not given permission, it is not appropriate to proceed with proposing to her.

If she accepted the suitor, then changed her mind, the ruling of acceptance is removed. Likewise, if her guardian changes his mind. But if neither she nor her guardian changes their mind, yet the suitor abandons the engagement or gives permission for it, then it is permissible to propose to her; based on the hadith of Ibn Umar: "He (the Prophet) forbade a man from proposing to a woman already engaged to his brother, unless he (the brother) gives him permission or leaves her."

If you fear that the suitor might disgrace his fiancée or harm her if you seek his permission, then it is better not to ask him.