Are the questioner and her sister considered disobedient to their parents due to the problems stemming from the parents' separation and the mother's objection to the father's parenting style and his treatment of them, and what should they do to overcome their fear of the father's threats to annul the engagement?

You are not wrong or disobedient to your father for your mother announcing the engagement in her home. Your father has no right to threaten you with breaking off the engagement, nor does he have the power to prevent you from marrying a suitable suitor. Entrust your affair to Allah, be careful not to fall short in your duties towards your parents, and combine honoring them and being good to them with wisdom, patience, gentle persuasion, and kindness. Indeed, Allah does not let the reward of the good-doers be lost.