What is the ruling on a man shaking hands with a woman who extends her hand to him, and what is the ruling on shaking hands with the husband of a maternal aunt or the husband of a paternal aunt?

For a man to be the husband of your paternal aunt (amma) or maternal aunt (khala) does not make him a mahram to you, so it is not permissible to shake hands with him. And if he extends his hand, never shake it, for shyness in this situation is blameworthy.