Is it permissible for a wife to shake hands with her young male cousins, or to speak with them, or to shake hands with elderly neighbors, or with an elderly man in her father's house? And is it considered unlawful to force a wife to wear the niqab?

It is not permissible for a wife to shake hands with a man who is a stranger to her, whether he is a relative or not, and whether he is old or not, as long as he is an adult. The husband must prevent her from doing so. As for covering the face, the more correct opinion is that it is obligatory. The husband can invite his wife to do so kindly, and if she does not comply, he can compel her to do so, and she must obey him.