Is Iddah (waiting period) incumbent on an old woman whose husband had died?
Upon the death of the husband, Iddah is incumbent on the wife regardless of their age. And Allah Knows Best.
What is the Islamic ruling on adoption, and what is the legal window through which a person can be kind to an orphan girl, attend to her needs and raise her as his own?
Kindness to an orphan entails a great reward from Allah, but adoption is prohibited and invalid. Therefore, it is permissible for the aforementioned person (for example) to make his brother`s wife breastfeed that girl so that he becomes her uncle by suckling provided that he informs her that she isn`t his biological daughter.
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.
Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.