Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
Is it permissible to go to the bathroom wearing a necklace that has the word "Allah" inscribed on it?
All Perfect Praie be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.
Anything that contains words pertaining to Allah, The Almighty, is desired to be left outside when going to the bathroom, or to be hidden for fear of losing it. And Allah Knows Best.
What is the ruling on the unification of Adhan?
All perfect praise be to AllahThe Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions. The unified Adhan is an act of Sunnah performed on behalf of a city's population. In addition, any Masjid that doesn’t have a Mua'than (The person who calls for prayers) and doesn`t raise the Adhan misses the reward of this great act. This is because Adhan should be raised for every group of people. However, the unified Adhan should be called in one Masjid and heard through all other Masjids. And Allah Knows Best.
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.