Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.
If a woman makes a vow to slaughter a sheep, and her husband is the one who buys it for her from his own money, and he says: "It is for you until you fulfill your vow with it," Is this permissible, or must she buy it herself from her own money?
If her husband gave her the sheep as a donation for the puprose of fulfilling the oath she made and was slaughtered by the wife or the husband on her behalf then the vow she made is fulfilled. And Allah Knows Best.
What is the best manner in distributing the meat of the Aqeeqah?
It is better to divide all of it as cooked meat amongst the poor and the needy. Dividing it into three portions is a Sunnah, but it is permissible to keep all of it provided that a portion, even a small one, is given to the poor and needy (one kilo for example).
Sometimes I receive a message containing a picture of someone desecrating the Noble Quran. The sender asks recipients to send that message to their contacts as an act of defending the Quran. What should I do when receiving such messages?
Praise be to Allah, the Lord of the Worlds.
You don`t have to circulate that message, because this isn`t the proper way to defend the Quran. And Allah The Almighty Knows Best.