Is a man obligated to pay an expiation of one dinar for having intercourse with his menstruating wife after he suspected her of menstruating, and can this expiation be given to his sick sister or his retired father?

The default is to believe the wife in her claim of menstruation. If the husband does not believe her, he is considered to have intentionally engaged in intercourse during menstruation, and therefore he must repent to Allah and pay expiation. There is no objection to giving the expiation to the needy, sick sister.