Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
If a woman becomes pure from menstruation before the Fajr Adhan in Ramadan, is she required to fast?
If a woman becomes pure (from menstrual period) before the Fajr Adhan, she must fast, as the impediment preventing her from fasting has been removed. The Sharia maxim in this regard states: "When the impediment is removed, the obligation returns."
She should then make the intention to fast before Fajr and perform ghusl (ritual purification) for prayer, whether before or after Fajr.
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.
Can a woman perform I‘tikaf in her home?
No, a woman's I‘tikaf is not valid if performed at home, because her home is not a mosque.