Is it permissible for the husband's brother to live with the wife in the same house during the husband's frequent absences and long travels, knowing that the wife is alone with her young children, and is the presence of the husband's brother in this situation considered forbidden for her?

It is not permissible for the husband's brother to live with the wife and minor children, due to the prohibition of such in the Sunnah, and because Satan flows through the son of Adam like blood, and there is no obedience to a created being in disobedience to the Creator. If possible, the house can be divided so that the husband's brother has a separate, independent section.