What is the right of the poor widow upon her wealthy brothers, and do they get double her share in inheritance despite her need and their wealth?

You have thought well of Allah. Inheritance is not about need; rather, it is according to what Allah the Almighty has clarified: "for the male, what is equal to the share of two females." If the sister is in need, then her brothers are obliged to spend on her, so her right is preserved. Spending on the needy is a great act of kindness for which the doer is rewarded.