Do ear and nose drops break the fast?
Yes, ear and nose drops break the fast because both the nose and ears are open passages to the body cavity.
What is the ruling on the ablution of one who touches women prohibited to him by a temporary prohibition (mahram bi-hurma mu'aqqata)?
It is prohibited to touch a woman who is temporarily prohibited (meaning one whom it becomes permissible to marry after the impediment is removed, such as the wife of a brother or paternal uncle). Touching her without a barrier invalidates ablution. And Allah the Almighty knows best.
Is it permissible for a wife to refuse to go to bed with her husband (for sexual intercourse)?
It isn`t permissible for her to do so unless for a sound reason.
What type of illness that permits breaking the fast in Ramadan?
It is an illness that, if one fasts, there is a fear it may lead to their death or cause unbearable hardship.