Is it permissible for me to skip congregational prayers—in rare cases—when I feel anxious about my mother and sister, due to the absence of a man with them in the house?

There is no harm in you praying at home if your fear for your family is justified, such as the spread of crime. A person's fear for their family is an excuse that permits abstaining from congregational prayer at the mosque. And when the fear subsides, be keen to pray in the mosque, for Allah Almighty says: "O you who have believed, indeed, among your wives and your children are enemies to you, so beware of them" (At-Taghabun: 14), meaning they should beware lest their family distract them from righteous deeds.