Is the Saum (Fasting) of someone who ate and drank forgetfully while offering fasting of oath expiation invalidated?
Whosoever eats, or drinks forgetfully is exempted by Allah; therefore, he/she shouldn`t break their fast whether it was obligatory, non-obligatory, or expiatory. And Allah Knows Best.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
Is it permissible for a wife to give her money to her family as a charity, or a gift without asking her husband, or seeking his consent?
The wife has the right to give her money as a charity, or a gift to her family, or to other people after consulting her husband out of respect, and this is the meaning of treating on footing of kindness and equity. Therefore, if he wanted to stop her from helping her family, then there is no harm in not telling him.
How are gaps filled during congregational prayer?
A praying person is allowed to take two steps in order to fill the gap in the row ahead. If the distance was long and requires much movement, then it is impermissible for one to move to fill it because much consecutive movement (three movements) during prayer invalidates it.