Is it permissible for a wife to inquire about her husband's professional affairs and his external relationships with people? And is it permissible for a man to visit a sick non-mahram woman in the hospital in the presence of her husband or one of her mahrams?

It is not the wife's right, Islamically, to pry into her husband's professional affairs and external relationships without his permission. Her right upon him is that he spends on her and treats her with kindness. It is permissible for a man to visit a non-mahram woman without seclusion if there is no fear of temptation.