What is the ruling on a person who broke the fast a day in Ramadan due to hardship?

He/she is obliged to make up that day.

What is the ruling on performing voluntary prayers during official working hours?

All perfect praise 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

There is no harm if  there was break time during official working hours, but if not then it is impermissible. And Allah Knows Best.

Is it permissible for a pious Muslim woman to stay with her husband who had quit praying out of laziness?

Quitting prayer is one of the major grave sins that come after disbelieving in Allah, but the wife of such a person isn`t considered divorced, rather, she should exert all her efforts in order to bring him back to the way of Islam. However, if his sin is likely to lure her from the way of Islam, then she had better separate from him by lawful means such as Mokhal`aa (When a wife pays a compensation for her husband in return for divorcing her). Moreover, if she exercises patience, and remains steadfast on her faith, then there is no harm in doing that.

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.