What is the ruling on the brother-in-law (sister's husband) staying in his sister's house when she is not present, accompanied by his children, and with only the questioner's wife and daughters present?

This action is permissible if certain conditions are met: there should be no suspicion regarding his arrival at the house before you, he should not know of your dislike for him entering your house in your absence, and his entry should not lead to any temptation. If any of these conditions are violated, then it is not permissible. It is better for him to wait until you go with him to the house; that is safer, further from suspicion, and more protective of honor.