I`m a pious Muslim woman, but my husband isn`t, what should I do?
You should exercise patience, make supplication that Allah guides him to the straight path, and keep advising him kindly.
Is it permissible for a woman to observe voluntary fast without her husband's permission?
A woman is not permitted to observe a voluntary fast (nafl) while her husband is present without his permission.
I broke my oath and am now obligated to pay the expiation for a broken oath (kaffārat al-yamīn). Would it be valid to fulfill this by inviting the poor and needy to a meal for them to eat from?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
The expiation for a broken oath (kaffārat al-yamīn) is given to the poor (fuqarā') and needy (masākīn), and it must be given in the form of ownership (tamlīk) transferred to them of the required amount — namely, one mudd of the staple food of the locality, or its equivalent monetary value. It isn`t valid to invite the poor or needy to a meal prepared for them to eat from, as this does not constitute "ownership" (tamlīk) but merely "permission to partake" (ibāḥah).
Imam al-Shirbīnī (may Allah have mercy on him) said: "What is intended is that ownership be transferred to them; therefore, providing them with lunch or dinner [as a meal to eat from] isn`t valid." [Summarized from Mughnī al-Muhtāj,Vol. 5/P.50]. And Allah the Almighty knows best.
What is the ruling on swearing on the Holy Quran and breaking that oath?
It is impermissible to swear on the Holy Quran in vain, and it is also impermissible to subject it to such an act since breaking the oath after swearing on the Quran entails an expiation.