Is it permissible for a wife to leave her house without asking her husband?
It is impermissible for the wife to leave her house without asking her husband except for a sound reason.
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.
Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
Can someone who begins a voluntary fast break it?
It is preferable for someone who begins an act of worship not to break it.
Allah the Exalted has said {what means}: "and let not your [good] deeds come to nought!" [Muhammad/33].
However, if a person starts a voluntary fast (nafl) and needs to break it, they are going against what is preferable, but there is no sin upon them.
