Is it permissible for my deceased brother's wife to come and stay with us in our house, knowing that she did not bear him any children and that I have grown sons, and what are her rights upon us after his death?

There is no harm in the deceased brother's wife staying overnight, provided that she does not display her adornment to your husband and children, and that she does not seclude herself with them, because she is a stranger to them. The deceased brother's wife does not have an obligatory right, but it is from the ethics of believers to maintain good relations and preserve affection.