Is it permissible to greet and sit with one's brother's widow, and for her to live with her husband's brothers in the same house, especially since she is not a mahram to them, is young, and has no mahram with her?

The wife of the brother is a stranger to her husband's brothers and is not an unmarriageable kin (mahram) to them, neither during his lifetime nor after his death. It is permissible to sit with her in the same place without seclusion (khalwa) if she observes proper Islamic حجاب (hijab). It is also permissible to speak with her out of necessity and to the extent of that necessity. However, it is not permissible for her to live with her husband's brothers in the same house unless she has separate private facilities (kitchen, bathroom, entrance, and exit) to avoid seclusion, unwarranted gazing, and exceeding the necessary extent in conversation.