What is the ruling on neighboring and eating the food of a female neighbor who works in a mixed profession, encompassing both permissible and prohibited elements, and who visits us and offers us food?

If the majority of your neighbor's wealth is lawful, then there is no harm in eating from her food. However, if the majority of her wealth is unlawful, then it is preferable to refrain from benefiting from it, and it is not permissible to benefit from the unlawful money itself. There is no harm in living next to her, as Islamic law enjoins good treatment of neighbors.