What is the ruling on entering the toilet with something containing the remembrance of Allah?

It is disliked to bring into the toilet anything containing a reminder of Allah, such as the Quran or a sacred name (like the name of Allah, Muhammad, etc.). If one does that, etiquette dictates that he conceal it so it is not visible while entering the toilet, either by cupping his hands over it, putting it in his pocket, or inside his shirt if it is a necklace or ring. And Allah  the Almighty 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.

Is it permissible to pay the Fitr Zakah (obligatory charity) of Ramadan on behalf of a dead person?

The Fitr Zakah of Ramadhaan isn`t due on one who had passed away before the sunset of the last day of Ramadan. And Allah Knows Best.

Do pregnant and breastfeeding women have to fast?